#6576503

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
prime radical	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2670 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)(7,12)(8,10)(9,11)(13,14,18,23)(15,20,27,38)(16,29,33,21)(17,34,31,22)(19,24,32,41)(25,43,44,36)(26,40,30,37)(28,39,35,42);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,30)(17,25)(18,27)(19,35)(21,40)(22,36)(23,38)(24,42)(26,33)(28,32)(29,37)(31,44)(34,43)(39,41);(2,4)(13,16)(14,21)(15,25)(17,32)(18,33)(19,31)(20,36)(22,41)(23,29)(24,34)(26,28)(27,44)(30,35)(37,39)(38,43)(40,42);(5,8,9,7)(6,10,11,12)(16,31)(17,33)(19,32)(21,34)(22,29)(24,41)(25,26)(28,35)(30,44)(36,37)(39,42)(40,43);(13,17,18,31)(14,22,23,34)(15,26,27,30)(16,32,33,19)(20,37,38,40)(21,41,29,24)(25,28,44,35)(36,39,43,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6483699);