#7699068

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

5222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606264);