#7518682

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

1944 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)(13,14,19,25)(15,27,31,21)(16,33,29,42)(17,37,28,43)(18,26,20,24)(22,36,41,35)(23,34,44,38)(30,39,32,40);(1,2)(3,4)(6,9)(10,11)(13,15,19,31)(14,21,25,27)(16,34,29,38)(17,35,28,36)(18,30,20,32)(22,37,41,43)(23,42,44,33)(24,39,26,40);(13,16,19,29)(14,22,25,41)(15,28,31,17)(18,36,20,35)(21,23,27,44)(24,42,26,33)(30,34,32,38)(37,40,43,39);(2,4)(13,17,19,28)(14,23,25,44)(15,29,31,16)(18,34,20,38)(21,22,27,41)(24,43,26,37)(30,36,32,35)(33,40,42,39);(5,7,8,12)(6,10,9,11)(13,18,19,20)(14,24,25,26)(15,30,31,32)(16,35,29,36)(17,34,28,38)(21,39,27,40)(22,33,41,42)(23,43,44,37)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,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=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, 7425878);