#8095362

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
prime radical	#35660	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002558);