#6438834

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3228 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346030);