#7704124

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4996 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)(3,8)(4,6)(5,7)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(9,11)(10,13)(12,16)(14,15)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32);(10,14)(11,16)(20,29)(23,28)(26,32)(30,31);(11,16)(13,15)(18,24)(22,25)(23,28)(30,31);(1,4,5,3)(2,6,7,8)

Polycyclic presentation:

2^9|b^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7611320);