#464659

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
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,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, 371855);