#7537888

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#45196	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1852 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7445084);