#6630001

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628847
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3116 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)(9,10,15,20)(11,16,21,29)(12,25,27,34)(13,19,14,18)(17,28,33,26)(22,36,35,40)(23,32,24,31)(30,38,39,37);(1,2,3,4)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,30)(18,31)(19,32)(20,29)(25,36)(26,37)(27,35)(28,38)(33,39)(34,40);(5,6,7,8)(11,21)(16,29)(22,35)(23,24)(30,39)(31,32)(36,40)(37,38);(9,12)(10,17)(11,22)(13,26)(14,28)(15,27)(16,30)(18,25)(19,34)(20,33)(21,35)(23,37)(24,38)(29,39)(31,36)(32,40);(9,13,15,14)(10,18,20,19)(11,23,21,24)(12,26,27,28)(16,31,29,32)(17,25,33,34)(22,37,35,38)(30,36,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537197);