#6575553

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2178 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,25,27,20)(16,21,17,22)(19,24,33,41)(26,43,44,35)(28,42,34,36)(29,39,31,37)(30,38,32,40);(6,9)(10,12)(13,15,18,27)(14,20,23,25)(16,29,17,31)(19,34,33,28)(21,37,22,39)(24,42,41,36)(26,32,44,30)(35,40,43,38);(2,4)(13,16,18,17)(14,21,23,22)(15,26,27,44)(19,32,33,30)(20,35,25,43)(24,40,41,38)(28,31,34,29)(36,39,42,37);(5,7,8,11)(6,10,9,12)(16,30)(17,32)(19,33)(21,38)(22,40)(24,41)(26,29)(28,34)(31,44)(35,37)(36,42)(39,43);(13,17,18,16)(14,22,23,21)(15,26,27,44)(19,30,33,32)(20,35,25,43)(24,38,41,40)(28,31,34,29)(36,39,42,37)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482749);