#7501206

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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2219 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)(7,8)(9,11)(10,12)(13,14,18,21)(15,19,26,32)(16,20,27,35)(17,22,29,36)(23,30,25,33)(24,31,38,43)(28,34,40,44)(37,41,39,42);(5,7)(6,10)(8,11)(9,12)(13,15,18,26)(14,19,21,32)(16,24,27,38)(17,23,29,25)(20,31,35,43)(22,30,36,33)(28,37,40,39)(34,41,44,42);(2,4)(5,8,9,10)(6,7,11,12)(15,23)(17,29)(19,30)(22,36)(24,37)(25,26)(28,40)(31,41)(32,33)(34,44)(38,39)(42,43);(2,4)(6,11)(7,12)(13,16)(14,20)(15,24)(17,28)(18,27)(19,31)(21,35)(22,34)(23,37)(25,39)(26,38)(29,40)(30,41)(32,43)(33,42)(36,44);(14,21)(16,27)(19,32)(22,36)(24,38)(28,40)(30,33)(37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,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, 7408402);