#6433081

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2728 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,25)(13,18)(14,29)(15,19)(16,23)(17,26)(21,30)(22,31)(24,35)(27,33)(28,32)(34,36)(37,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,15,24,14,23)(10,16,29,35,19,31,18,20)(12,21,26,36,28,38,27,37)(17,30,25,39,33,40,32,34);(3,7)(5,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,28)(16,30)(18,32)(19,33)(20,34)(22,36)(23,37)(24,38)(25,29)(31,40)(35,39);(2,6)(5,8)(9,10)(11,16)(12,17)(13,29)(14,18)(15,19)(20,23)(21,30)(22,35)(24,31)(25,26)(27,32)(28,33)(34,37)(36,39)(38,40);(9,13,15,14)(10,18,19,29)(11,22,24,23)(12,26,28,27)(16,20,31,35)(17,32,33,25)(21,36,38,37)(30,34,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340277);