#7515403

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	#25125	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1971 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)(13,14,18,23)(15,25,28,20)(16,30,32,39)(17,24,19,21)(22,34,40,33)(26,36,29,37)(27,42,44,38)(31,41,43,35);(1,2)(3,4)(6,9)(10,12)(13,15,18,28)(14,20,23,25)(16,31,32,43)(17,29,19,26)(21,37,24,36)(22,38,40,42)(27,34,44,33)(30,35,39,41);(5,7,8,11)(6,10,9,12)(14,21)(15,26)(16,32)(17,19)(22,39)(23,24)(27,43)(28,29)(30,40)(31,44)(35,41)(36,37);(2,4)(13,16,18,32)(14,22,23,40)(15,27,28,44)(17,33,19,34)(20,35,25,41)(21,30,24,39)(26,31,29,43)(36,38,37,42);(6,9)(10,12)(13,17,18,19)(14,21,23,24)(15,26,28,29)(16,33,32,34)(20,36,25,37)(22,30,40,39)(27,31,44,43)(35,38,41,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=i,e^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,e^b=ei,g^b=gi,h^b=hi,d^c=di,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, 7422599);