#422826

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fi,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, 330022);