#7514998

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	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1973 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)(15,23)(16,27)(17,30)(19,24)(20,28)(21,31)(26,32)(29,33)(35,37)(36,38)(39,42)(40,44);(1,2)(3,4)(6,9)(10,12)(13,14,18,22)(15,24,26,37)(16,28,29,38)(17,31,30,21)(19,32,35,23)(20,33,36,27)(25,34,41,43)(39,44,42,40);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,25)(17,32)(18,26)(20,34)(21,37)(22,35)(23,30)(24,31)(27,39)(28,40)(29,41)(33,42)(36,43)(38,44);(2,4)(13,16,18,29)(14,20,22,36)(15,25,26,41)(17,33,30,27)(19,34,35,43)(21,38,31,28)(23,39,32,42)(24,40,37,44);(6,9)(10,12)(15,26)(19,35)(23,32)(24,37)(25,41)(34,43)(39,42)(40,44)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,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, 7422194);