#6617648

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2995 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)(13,14,19,31,18,22,17,23)(15,24,30,20,27,37,26,34)(16,21,32,42,28,35,29,36)(25,38,41,33,39,44,40,43);(6,10)(8,11)(13,15,18,27)(14,20,22,34)(16,25,28,39)(17,30,19,26)(21,33,35,43)(23,37,31,24)(29,41,32,40)(36,44,42,38);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16,18,28)(14,21,22,35)(15,25,27,39)(17,29,19,32)(20,33,34,43)(23,36,31,42)(24,38,37,44)(26,40,30,41);(5,7,9,12)(6,11,10,8)(16,28)(21,35)(25,39)(29,32)(33,43)(36,42)(38,44)(40,41);(5,8,9,11)(6,7,10,12)(14,22)(20,34)(21,35)(23,31)(24,37)(33,43)(36,42)(38,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,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, 6524844);