#7505967

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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2334 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)(17,18,21,22)(19,25,27,23)(20,24,29,30)(26,28,32,31);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,19,21,27)(18,23,22,25)(20,28,29,31)(24,26,30,32);(10,14)(13,16)(17,20,21,29)(18,24,22,30)(19,26,27,32)(23,31,25,28);(2,6)(3,8)(10,14)(13,16)(17,19)(18,25)(20,26)(21,27)(22,23)(24,28)(29,32)(30,31);(19,27)(23,25)(26,32)(28,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,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, 7413163);