#7633224

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

5142 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540420);