#147802

order	512
nilpotency class	4
isoclinism	isoclinic class of #13350
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10995
prime radical	#60749	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1024 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1024 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,e^d=ei,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54998);