#395679

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^c=ei,g^c=gi

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, 302875);