#7565500

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

1786 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)(13,14,20,26)(15,21,29,23)(16,30,33,41)(17,36,28,38)(18,25,19,24)(22,35,40,34)(27,39,43,44)(31,37,32,42);(1,2)(3,4)(6,10)(11,12)(13,15,20,29)(14,21,26,23)(16,31,33,32)(17,18,28,19)(22,39,40,44)(24,36,25,38)(27,35,43,34)(30,37,41,42);(5,7,9,8)(6,11,10,12)(13,16,18,34,20,33,19,35)(14,22,24,30,26,40,25,41)(15,27,17,31,29,43,28,32)(21,37,38,39,23,42,36,44);(2,4)(5,7,9,8)(6,11,10,12)(13,17,19,15,20,28,18,29)(14,23,25,38,26,21,24,36)(16,32,35,43,33,31,34,27)(22,37,41,44,40,42,30,39);(5,8,9,7)(6,11,10,12)(16,33)(22,40)(27,43)(30,41)(31,32)(34,35)(37,42)(39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fgi,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,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7472696);