#6630261

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2162 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,30)(17,22)(18,24)(19,25)(20,34)(23,33)(26,39)(27,37)(28,38)(29,43)(31,35)(32,40)(36,41)(42,44);(6,9)(10,12)(13,15)(14,21)(16,31)(17,26)(18,27)(19,28)(20,29)(22,39)(23,36)(24,37)(25,38)(30,35)(32,42)(33,41)(34,43)(40,44);(1,2,3,4)(13,16,19,17)(14,22,25,30)(15,26,28,31)(18,33,20,32)(21,35,38,39)(23,24,40,34)(27,42,29,41)(36,43,44,37);(13,17,19,16)(14,23,25,40)(15,26,28,31)(18,33,20,32)(21,36,38,44)(22,24,30,34)(27,41,29,42)(35,43,39,37);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,24,25,34)(15,27,28,29)(16,32,17,33)(21,37,38,43)(22,23,30,40)(26,41,31,42)(35,44,39,36)

Polycyclic presentation:

2^9|c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,c^b=ci,e^c=ei,h^d=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537457);