#7501148

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500740
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,21)(12,24)(13,28)(14,20)(15,19)(16,27)(17,30)(18,33)(22,38)(23,26)(25,39)(29,40)(31,37)(32,36)(34,35);(1,3,4,8)(2,5,6,7)(9,11)(10,12)(13,29)(14,16)(15,24)(17,35)(18,27)(19,21)(20,26)(22,31)(23,33)(25,36)(28,38)(30,39)(32,34)(37,40);(9,12)(10,16)(11,18)(13,25)(14,26)(15,21)(17,34)(19,23)(20,24)(22,28)(27,33)(29,37)(30,38)(31,35)(32,40)(36,39);(2,6)(3,8)(9,13)(10,17)(11,22)(12,25)(14,30)(15,31)(16,34)(18,28)(19,36)(20,37)(21,35)(23,39)(24,29)(26,38)(27,40)(32,33);(9,14,15,33)(10,18,19,20)(11,23,24,16)(12,26,21,27)(13,30,31,32)(17,28,36,37)(22,39,29,34)(25,38,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228033, #72228038
maximal normal subgroup of	#72228033, #72228038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408344);