#7503808

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2136 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7411004);