#6558270

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2497 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465466);