#7503556

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

2018 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)(16,28)(17,31)(18,23)(19,22)(21,30)(24,41)(25,44)(26,38)(27,37)(29,39)(32,40)(33,36)(34,43)(35,42);(5,7,9,6)(8,11,12,10)(13,15,18,27)(14,20,23,37)(16,25,30,42)(17,26,19,33)(21,35,28,44)(22,36,31,38)(24,32,43,29)(34,39,41,40);(2,4)(5,8,9,12)(6,10,7,11)(13,16)(14,21)(15,24)(17,29)(18,30)(19,32)(20,34)(22,39)(23,28)(25,33)(26,42)(27,43)(31,40)(35,36)(37,41)(38,44);(2,4)(6,7)(10,11)(13,16)(14,21)(15,25)(17,32)(18,30)(19,29)(20,35)(22,40)(23,28)(24,33)(26,43)(27,42)(31,39)(34,36)(37,44)(38,41);(13,17,18,19)(14,22,23,31)(15,26,27,33)(16,29,30,32)(20,36,37,38)(21,39,28,40)(24,42,43,25)(34,35,41,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,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,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, 7410752);