#419007

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2253 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)(5,6,10,14,11,16,9,15)(7,17,21,28,22,18,20,12)(8,13,24,31,26,32,25,30)(19,33,23,36,34,29,35,27);(5,7,9,20,11,22,10,21)(6,12,14,17,16,28,15,18)(8,23,24,34,26,35,25,19)(13,29,30,36,32,33,31,27);(5,6)(7,18)(8,13)(9,14)(10,15)(11,16)(12,20)(17,22)(19,29)(21,28)(23,36)(24,30)(25,31)(26,32)(27,35)(33,34);(2,4)(5,8,11,26)(6,13,16,32)(7,19,22,34)(9,25,10,24)(12,27,28,36)(14,31,15,30)(17,33,18,29)(20,35,21,23)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,f^c=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, 326203);