#7675390

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

4918 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7582586);