#7501750

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2367 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:

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

Polycyclic presentation:

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

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, 7408946);