#7503558

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2056 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:

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

Polycyclic presentation:

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

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, 7410754);