#6498002

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470362
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,33)(28,37)(29,31)(30,36)(32,34)(35,39)(38,40);(2,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,34,40,29,35,28,36)(26,30,37,39,31,38,32,33);(1,3,4,8)(2,6,5,7)(11,17)(14,21)(18,23)(22,24)(26,31)(30,38)(32,37)(33,39);(11,17)(14,21)(18,23)(22,24);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,35)(30,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=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, 6405198);