#455156

order	512
nilpotency class	3
isoclinism	isoclinic class of #455155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

1542 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362352);