#6519848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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	#35212	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1789 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh

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, 6427044);