#7503574

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 7410770);