#7713424

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

4953 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)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,28)(21,29)(22,24)(25,30)(26,32)(27,31);(1,2)(3,7)(4,8)(5,6)(17,19,22,25)(18,23,24,30)(20,26,29,31)(21,27,28,32);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(18,24)(19,25)(20,29)(27,32);(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,21)(19,26)(22,29)(23,27)(24,28)(25,31)(30,32);(1,4,5,3)(2,8,6,7)(17,21)(18,20)(19,27)(22,28)(23,26)(24,29)(25,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7620620);