#422827

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	#23426	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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,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, 330023);