#7876851

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

4992 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=hi,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7784047);