#6442926

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2660 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:

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

Polycyclic presentation:

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

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, 6350122);