#7631999

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

5118 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)(3,8)(9,10,13,16)(11,14,12,15)(17,18)(19,22)(20,26)(21,27)(23,28)(24,31)(25,30)(29,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,31)(25,32)(26,27);(1,3,5,8)(2,7,6,4)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,21)(22,32)(23,30)(24,27)(26,28)(29,31);(9,11,13,12)(10,14,16,15)(17,20)(18,26)(19,29)(21,31)(22,32)(23,30)(24,27)(25,28);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,15,16,14)(17,21,23,29)(18,27,28,32)(19,30,31,20)(22,25,24,26);(17,22)(18,19)(20,32)(21,25)(23,24)(26,29)(27,30)(28,31)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=gh,e^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539195);