#8721201

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

5222 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)(12,16)(17,18,22,23)(19,26,24,20)(21,25,31,32)(27,30,29,28);(1,2)(3,4)(5,7)(6,8)(9,10,14,13)(11,16,15,12)(18,23)(19,24)(25,32)(27,29);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(18,23)(19,24)(25,32)(27,29);(3,8)(4,6)(10,13)(11,15)(17,19,22,24)(18,20,23,26)(21,29,31,27)(25,30,32,28);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32);(1,4,5,6)(2,3,7,8)(9,13,14,10)(11,16,15,12)(17,21,22,31)(18,25,23,32)(19,27,24,29)(20,28,26,30)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8628397);