#422892

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1933 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^d=ei,h^d=hi

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, 330088);