#6567592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#28452	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1879 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6474788);