#6442906

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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

2624 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)(11,20)(12,24)(13,18)(14,19)(15,29)(16,23)(17,27)(21,35)(22,32)(25,39)(26,33)(28,31)(30,40)(34,37)(36,38);(2,4)(5,6)(7,8)(9,11)(10,16)(12,21)(13,28)(14,23)(15,22)(17,30)(18,32)(19,20)(24,38)(25,36)(26,40)(27,37)(29,31)(33,34)(35,39);(6,8)(9,12,15,27,14,25,13,26)(10,17,18,24,19,33,29,39)(11,21,22,37,23,36,28,40)(16,30,32,38,20,34,31,35);(6,8)(12,25)(17,33)(21,36)(24,39)(26,27)(30,34)(35,38)(37,40);(9,13,14,15)(10,18,19,29)(11,22,23,28)(12,26,25,27)(16,31,20,32)(17,24,33,39)(21,37,36,40)(30,35,34,38)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,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, 6350102);