#7501102

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6,8,10)(7,11,12,9)(13,14)(15,23)(16,27)(17,22)(18,31)(19,25)(20,29)(21,30)(24,39)(26,35)(28,36)(32,37)(33,43)(34,42)(38,41)(40,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,24)(17,25)(18,26)(20,33)(21,35)(22,23)(27,44)(28,40)(29,41)(30,42)(31,34)(32,43)(36,39)(37,38);(2,4)(13,16)(14,20)(15,24)(17,28)(18,32)(19,33)(21,36)(22,37)(23,38)(25,40)(26,43)(27,31)(29,30)(34,44)(35,39)(41,42);(6,10)(7,12)(14,21)(15,25)(18,30)(19,34)(20,36)(22,31)(23,35)(24,40)(27,37)(29,32)(33,44)(38,39);(14,22)(16,28)(19,23)(21,31)(24,40)(29,32)(34,35)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408298);