#464658

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

1528 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)(5,6,10,15)(7,16,19,12)(8,13,22,32)(9,24,26,14)(11,25,28,18)(17,29,35,34)(20,27,36,30)(21,33,23,31);(5,7,9,18,10,19,26,25)(6,11,14,16,15,28,24,12)(8,17,23,36,22,35,21,20)(13,27,33,34,32,30,31,29);(2,4)(5,7)(6,12)(8,20)(9,25)(10,19)(11,24)(13,30)(14,28)(15,16)(17,21)(18,26)(22,36)(23,35)(27,32)(29,31)(33,34);(5,8)(6,13)(7,17)(9,23)(10,22)(11,27)(12,29)(14,33)(15,32)(16,34)(18,36)(19,35)(20,25)(21,26)(24,31)(28,30)

Polycyclic presentation:

2^9|a^2=f,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, 371854);