#500258

order	512
nilpotency class	4
isoclinism	isoclinic class of #500256
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24058
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3485	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

819 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 819 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,e^b=ei,f^b=fh,d^c=dh,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407454);