#6443106

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5066 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)(9,10)(11,20)(12,17)(13,18)(14,28)(15,19)(16,22)(21,34)(23,30)(24,32)(25,33)(26,38)(27,31)(29,36)(35,40)(37,39);(5,6)(7,8)(9,11,14,22,15,23,13,27)(10,16,18,20,19,31,28,30)(12,21,26,36,24,35,25,37)(17,29,33,34,32,39,38,40);(1,2,3,4)(6,8)(9,12,15,24)(10,17,19,32)(11,21,23,35)(13,25,14,26)(16,29,31,39)(18,33,28,38)(20,34,30,40)(22,36,27,37);(6,8)(12,24)(17,32)(21,35)(25,26)(29,39)(33,38)(34,40)(36,37);(9,13,15,14)(10,18,19,28)(11,22,23,27)(12,25,24,26)(16,30,31,20)(17,33,32,38)(21,36,35,37)(29,40,39,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350302);