#8664532

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	#59003	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4920 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571728);