#7655225

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

5693 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7562421);