#7565189

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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

1779 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)(8,12)(9,10)(13,14,19,24)(15,26,29,21)(16,30,34,41)(17,36,28,38)(18,25,20,23)(22,35,40,33)(27,39,43,44)(31,37,32,42);(1,2)(3,4)(6,10)(11,12)(13,15,19,29)(14,21,24,26)(16,31,34,32)(17,18,28,20)(22,39,40,44)(23,38,25,36)(27,33,43,35)(30,42,41,37);(5,7,9,8)(6,11,10,12)(13,16,20,35,19,34,18,33)(14,22,25,41,24,40,23,30)(15,27,28,31,29,43,17,32)(21,37,38,39,26,42,36,44);(2,4)(5,7,9,8)(6,11,10,12)(13,17,20,15,19,28,18,29)(14,21,25,38,24,26,23,36)(16,32,35,27,34,31,33,43)(22,37,41,39,40,42,30,44);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,23,24,25)(15,28,29,17)(16,33,34,35)(21,38,26,36)(22,30,40,41)(27,31,43,32)(37,39,42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fg,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7472385);