#7711327

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

5358 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618523);