#6442942

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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

2592 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)(15,30)(16,23)(17,26)(21,37)(22,33)(25,40)(27,35)(28,32)(29,36)(31,38)(34,39);(2,4)(5,6)(7,8)(9,11)(10,16)(12,25)(13,28)(14,23)(15,22)(17,34)(18,20)(19,33)(21,27)(24,37)(26,38)(29,39)(30,32)(31,36)(35,40);(6,8)(9,12,15,26,14,27,13,29)(10,17,19,24,20,36,30,35)(11,21,22,39,23,25,28,38)(16,31,33,40,18,34,32,37);(6,8)(9,11)(10,18)(12,21)(13,28)(14,23)(15,22)(16,20)(17,34)(19,32)(24,37)(25,27)(26,39)(29,38)(30,33)(31,36)(35,40);(9,13,14,15)(10,19,20,30)(11,22,23,28)(12,26,27,29)(16,32,18,33)(17,35,36,24)(21,38,25,39)(31,40,34,37)

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=dhi,e^b=ei,e^c=ei,e^d=ei

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, 6350138);