#7520437

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1844 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)(7,11)(8,10)(9,12)(13,14,19,25)(15,27,29,21)(16,30,33,43)(17,35,36,42)(18,26,20,24)(22,32,40,34)(23,38,39,31)(28,41,37,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,31)(17,34)(18,28)(19,29)(20,37)(22,42)(23,43)(24,41)(25,27)(26,44)(30,39)(32,36)(33,38)(35,40);(5,7,9,8)(6,11,12,10)(13,16,19,33)(14,22,25,40)(15,17,29,36)(18,34,20,32)(21,39,27,23)(24,30,26,43)(28,38,37,31)(35,44,42,41);(2,4)(13,17,19,36)(14,23,25,39)(15,16,29,33)(18,31,20,38)(21,40,27,22)(24,35,26,42)(28,32,37,34)(30,44,43,41);(5,8,9,7)(6,11,12,10)(13,18,19,20)(14,24,25,26)(15,28,29,37)(16,32,33,34)(17,31,36,38)(21,41,27,44)(22,43,40,30)(23,35,39,42)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7427633);