#7784338

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

5170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7691534);