#6630007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^b=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, 6537203);