#7702522

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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

4891 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)(3,7)(4,8)(5,6)(10,12)(11,15)(17,18)(19,26)(20,25)(21,30)(22,23)(24,28)(27,32)(29,31);(2,6)(7,8)(9,10,13,12)(11,16,15,14)(18,23)(24,26)(25,30)(31,32);(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,26)(25,32)(30,31);(1,3,5,4)(2,7,6,8)(9,10,13,12)(11,14,15,16)(17,20,22,21)(18,25,23,30)(19,27,28,29)(24,31,26,32);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,14,15,16)(17,21,22,20)(18,25,23,30)(19,27,28,29)(24,32,26,31);(1,4,5,3)(2,8,6,7)

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,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, 7609718);