#464064

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1519 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)(7,13)(8,14)(9,16)(10,15)(11,17)(12,19)(18,33)(20,28)(21,27)(22,26)(23,32)(24,31)(25,30)(29,34)(35,36);(5,7,10,20,11,21,9,19)(6,12,16,27,17,28,15,13)(8,18,23,22,24,34,25,35)(14,26,30,33,31,36,32,29);(2,4)(5,7)(6,13)(8,22)(9,20)(10,19)(11,21)(12,15)(14,26)(16,28)(17,27)(18,23)(24,35)(25,34)(29,30)(31,36)(32,33);(5,8)(6,14)(7,18)(9,25)(10,23)(11,24)(12,26)(13,29)(15,32)(16,30)(17,31)(19,35)(20,22)(21,34)(27,33)(28,36)

Polycyclic presentation:

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

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, 371260);