#7744736

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

4995 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:

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

Polycyclic presentation:

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

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, 7651932);