#8395039

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8302235);