#427978

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 335174);