#7613542

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#45385	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1975 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)(7,10)(8,9)(11,12)(15,25)(16,30)(17,18)(20,26)(21,38)(22,23)(27,32)(28,29)(31,40)(33,34)(35,39)(36,37);(2,4)(6,9)(10,12)(13,14,19,24)(15,26,28,37)(16,31,34,38)(17,23,18,22)(20,29,36,25)(21,30,40,33)(27,35,32,39)(41,44,43,42);(5,7,8,11)(6,10,9,12)(13,15,18,29,19,28,17,25)(14,20,23,37,24,36,22,26)(16,32,33,41,34,27,30,43)(21,39,31,42,40,35,38,44);(1,2,3,4)(13,16,17,30,19,34,18,33)(14,21,22,38,24,40,23,31)(15,27,25,41,28,32,29,43)(20,35,26,42,36,39,37,44);(6,9)(10,12)(15,28)(20,36)(25,29)(26,37)(27,32)(35,39)(41,43)(42,44)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=hi,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,d^c=di,e^c=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, 7520738);