#511933

order	512
nilpotency class	4
isoclinism	isoclinic class of #511841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#32926
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1023 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,13)(4,18)(5,23)(6,11)(7,12)(8,16)(9,22)(10,25)(14,38)(15,36)(17,37)(19,48)(20,46)(21,52)(24,55)(26,34)(27,57)(28,42)(29,44)(30,60)(31,51)(32,62)(33,56)(35,41)(39,47)(40,58)(43,63)(45,49)(50,61)(53,54)(59,64);(1,3,5,16,7,15,25,42)(2,8,10,13,12,28,23,36)(4,19,21,49,20,14,39,40)(6,17,24,44,26,43,56,64)(9,30,32,61,31,27,54,35)(11,29,33,37,34,59,55,63)(18,45,47,48,46,58,52,38)(22,50,53,60,51,41,62,57);(1,4)(2,9)(3,14)(5,21)(6,22)(7,20)(8,27)(10,32)(11,18)(12,31)(13,35)(15,19)(16,40)(17,41)(23,54)(24,53)(25,39)(26,51)(28,30)(29,58)(33,47)(34,46)(36,61)(37,38)(42,49)(43,50)(44,57)(45,59)(48,63)(52,55)(56,62)(60,64);(3,15)(4,20)(6,26)(8,28)(9,31)(13,36)(16,42)(18,46)(21,39)(24,56)(29,59)(32,54)(37,63)(41,50)(47,52)(57,60)

Polycyclic presentation:

2^9|b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,c^b=ci,d^b=di,d^c=di,g^c=ghi,f^d=fh,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303788, #72303804, #72303806, …
maximal normal subgroup of	#72303788, #72303804, #72303806, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419129);