#7971698

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

5024 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,12,15)(17,18,21,22)(19,25,26,23)(20,24,29,28)(27,31,32,30);(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(18,22)(19,26)(24,28)(27,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31);(1,3)(2,6)(4,8)(5,7)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32);(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7878894);