#7904712

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

4704 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7811908);