#424291

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1555 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:

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

Polycyclic presentation:

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

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, 331487);