#7501757

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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×#3041	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2485 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)(7,9)(8,10)(11,12)(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)(5,7,9,6)(8,11,12,10)(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)(5,8,9,12)(6,10,7,11)(13,16,17,28)(14,20,22,37)(15,24,25,40)(18,32,30,29)(19,33,23,38)(21,36,31,27)(26,43,42,41)(34,44,35,39);(6,7)(10,11)(14,21)(15,25)(18,30)(19,34)(20,36)(22,31)(23,35)(24,40)(27,37)(29,32)(33,44)(38,39);(16,28)(20,37)(24,40)(27,36)(29,32)(33,38)(39,44)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408953);