#7501190

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	#35150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2265 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)(7,8)(9,11)(10,12)(13,14)(15,20)(16,21)(17,22)(18,24)(19,23)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(41,43)(42,44);(5,7)(6,10)(8,11)(9,12)(13,15,19,27)(14,20,23,35)(16,28,30,41)(17,26,18,31)(21,36,38,43)(22,34,24,39)(25,32,42,29)(33,40,44,37);(2,4)(5,8,9,10)(6,7,11,12)(13,16,19,30)(14,21,23,38)(15,25,27,42)(17,29,18,32)(20,33,35,44)(22,37,24,40)(26,41,31,28)(34,43,39,36);(2,4)(6,11)(7,12)(13,17,19,18)(14,22,23,24)(15,26,27,31)(16,29,30,32)(20,34,35,39)(21,37,38,40)(25,41,42,28)(33,43,44,36);(14,23)(20,35)(21,38)(22,24)(33,44)(34,39)(36,43)(37,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^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

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, 7408386);