#6676050

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1822 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)(5,6,9,11)(7,12,8,10);(5,6)(7,10)(8,12)(9,11)(13,14,18,22)(15,23,26,34)(16,20,29,38)(17,30,32,21)(19,31,35,25)(24,39,27,41)(28,43,44,37)(33,42,36,40);(2,4)(13,15,18,26)(14,19,22,35)(16,27,29,24)(17,31,32,25)(20,36,38,33)(21,34,30,23)(28,40,44,42)(37,39,43,41);(5,7,9,8)(6,10,11,12)(13,16,18,29)(14,20,22,38)(15,24,26,27)(17,28,32,44)(19,33,35,36)(21,37,30,43)(23,39,34,41)(25,40,31,42);(5,8,9,7)(6,10,11,12)(14,21)(17,32)(19,34)(20,37)(22,30)(23,35)(25,31)(28,44)(33,41)(36,39)(38,43)(40,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6583246);