#7501216

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

2270 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)(13,14,19,24)(15,20,29,37)(16,21,31,39)(17,22,18,23)(25,33,41,43)(26,34,42,44)(27,35,28,36)(30,38,32,40);(5,7)(6,10)(8,11)(9,12)(13,15,19,29)(14,20,24,37)(16,26,31,42)(17,28,18,27)(21,34,39,44)(22,36,23,35)(25,32,41,30)(33,40,43,38);(2,4)(5,8,9,10)(6,7,11,12)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,33)(22,38)(23,40)(24,39)(26,27)(28,42)(29,41)(34,35)(36,44)(37,43);(2,4)(6,11)(7,12)(13,16)(14,21)(15,26)(17,32)(18,30)(19,31)(20,34)(22,40)(23,38)(24,39)(25,27)(28,41)(29,42)(33,35)(36,43)(37,44);(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,30,31,32)(20,35,37,36)(21,38,39,40)(25,26,41,42)(33,34,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,e^2=i,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, 7408412);