#422835

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|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,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330031);