#7613945

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1827 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)(15,25)(16,30)(17,18)(20,34)(21,37)(22,23)(26,36)(27,41)(28,29)(31,38)(32,33)(35,42);(2,4)(6,9)(10,11)(13,14,19,24)(15,26,29,34)(16,31,32,37)(17,23,18,22)(20,25,36,28)(21,30,38,33)(27,42,41,35)(39,44,43,40);(13,15,17,25,19,29,18,28)(14,20,22,34,24,36,23,26)(16,27,30,39,32,41,33,43)(21,35,37,44,38,42,31,40);(1,2,3,4)(13,16,17,30,19,32,18,33)(14,21,22,37,24,38,23,31)(15,27,25,39,29,41,28,43)(20,35,34,44,36,42,26,40);(5,7,8,12)(6,10,9,11)(16,32)(21,38)(27,41)(30,33)(31,37)(35,42)(39,43)(40,44)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=hi,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

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, 7521141);