#500026

order	512
nilpotency class	4
isoclinism	isoclinic class of #499867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#32890	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,21)(12,19)(13,16)(14,18)(15,29)(17,23)(20,26)(22,38)(24,35)(25,34)(27,31)(28,36)(30,37)(32,40)(33,39);(1,3,4,8)(2,5,6,7)(10,16)(11,22)(15,26)(17,32)(18,19)(21,37)(23,39)(24,30)(25,28)(27,40)(31,33)(35,38);(9,11)(10,17)(12,23)(13,24)(14,27)(15,30)(16,31)(18,21)(19,35)(20,32)(22,26)(25,40)(28,39)(29,33)(34,37)(36,38);(2,6)(5,7)(9,12,13,14)(10,18,16,19)(11,23,24,27)(15,25,26,28)(17,21,31,35)(20,34,29,36)(22,39,30,40)(32,37,33,38)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340658, #72340704, #72340710, …
maximal normal subgroup of	#72340658, #72340704, #72340710, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407222);