#7504373

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	#35527	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,e^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, 7411569);