#7915511

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

5381 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^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, 7822707);