#6598253

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#38903	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 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)(13,14,19,24)(15,25,29,35)(16,30,32,37)(17,22,18,23)(20,28,36,27)(21,33,38,31)(26,34,41,44)(39,42,43,40);(5,6,8,9)(7,10,11,12)(13,15,18,28,19,29,17,27)(14,20,22,35,24,36,23,25)(16,26,31,40,32,41,33,42)(21,34,30,43,38,44,37,39);(2,4)(6,9)(7,11)(13,16,17,33,19,32,18,31)(14,21,23,37,24,38,22,30)(15,26,27,42,29,41,28,40)(20,34,25,39,36,44,35,43);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,32,33)(20,35,36,25)(21,37,38,30)(26,40,41,42)(34,39,44,43);(6,9)(7,11)(16,32)(21,38)(26,41)(30,37)(31,33)(34,44)(39,43)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei,e^d=eh

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, 6505449);