#7590505

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497701);