#7518747

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1951 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)(13,14,18,24)(15,27,30,21)(16,33,29,42)(17,37,28,44)(19,26,20,25)(22,36,41,35)(23,34,43,38)(31,39,32,40);(1,2)(3,4)(6,9)(10,11)(13,15,18,30)(14,21,24,27)(16,34,29,38)(17,35,28,36)(19,32,20,31)(22,37,41,44)(23,42,43,33)(25,40,26,39);(13,16,18,29)(14,22,24,41)(15,28,30,17)(19,36,20,35)(21,23,27,43)(25,42,26,33)(31,38,32,34)(37,39,44,40);(2,4)(13,17,18,28)(14,23,24,43)(15,29,30,16)(19,34,20,38)(21,22,27,41)(25,44,26,37)(31,35,32,36)(33,39,42,40);(5,7,8,12)(6,10,9,11)(14,24)(15,30)(16,29)(23,43)(25,26)(31,32)(35,36)(37,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,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=cg,d^b=dh,e^b=ei,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, 7425943);