#6442938

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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6350134);