#7512323

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1554 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7419519);