#7971695

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

5040 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)(10,13)(14,16)(17,18)(19,28)(20,30)(21,32)(22,27)(23,25)(24,29)(26,31);(9,10,12,13)(11,14,15,16)(17,19,23,29)(18,24,25,28)(20,22,31,32)(21,26,27,30);(2,5)(6,7)(9,11,12,15)(10,14,13,16)(18,25)(19,29)(20,31)(21,27);(1,2)(3,6)(4,5)(7,8)(17,20)(18,26)(19,22)(21,28)(23,31)(24,27)(25,30)(29,32);(1,3)(2,6)(4,8)(5,7)(17,21)(18,22)(19,26)(20,28)(23,27)(24,31)(25,32)(29,30);(17,22)(18,27)(19,20)(21,25)(23,32)(24,26)(28,30)(29,31)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,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, 7878891);