#232901

order	512
nilpotency class	3
isoclinism	isoclinic class of #232897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60962	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1281 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)(3,5)(4,6)(7,8)(9,10,15,18)(11,21,24,35)(12,25,26,33)(13,20,14,19)(16,29,31,23)(17,27,34,28)(22,30,37,32)(36,39,40,38);(2,5)(3,8)(6,7)(9,11)(10,16)(12,22)(13,29)(14,23)(15,24)(17,32)(18,31)(19,35)(20,21)(25,36)(26,37)(27,39)(28,38)(30,34)(33,40);(1,2)(3,7)(4,6)(5,8)(9,12)(10,17)(11,22)(13,28)(14,27)(15,26)(16,30)(18,34)(19,25)(20,33)(21,36)(23,39)(24,37)(29,38)(31,32)(35,40);(2,6)(5,7)(10,18)(12,26)(16,31)(19,20)(21,35)(22,37)(27,28)(38,39)

Polycyclic presentation:

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

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, 140097);