#7904711

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

4872 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,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, 7811907);