#8697562

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	#36362	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5212 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=gi,e^2=h,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, 8604758);