#8699270

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	#36125	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4819 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8606466);