#7562529

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2206 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)(8,9)(10,12)(13,14)(15,25)(16,30)(17,20)(18,24)(19,23)(21,29)(22,34)(26,35)(27,39)(28,38)(31,36)(32,37)(33,40)(41,43)(42,44);(1,2)(3,4)(6,9)(10,11)(14,20)(15,26)(16,31)(18,19)(21,25)(22,30)(27,32)(28,29)(33,34)(35,38)(36,40)(41,44);(13,15,18,29,17,28,19,26)(14,21,23,25,20,35,24,38)(16,32,34,42,33,27,31,43)(22,39,30,44,36,37,40,41);(2,4)(13,16,18,34,17,33,19,31)(14,22,23,30,20,36,24,40)(15,27,29,43,28,32,26,42)(21,37,25,41,35,39,38,44);(5,7,8,12)(6,10,9,11)(14,20)(16,33)(21,35)(23,24)(25,38)(27,32)(31,34)(42,43)

Polycyclic presentation:

2^9|c^2=g,d^2=g,e^2=fi,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,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, 7469725);