#471457

order	512
nilpotency class	3
isoclinism	isoclinic class of #456884
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1136 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)(6,11)(9,23)(12,16)(13,30)(14,15)(17,33)(18,24)(21,25)(26,32)(27,31)(28,29)(35,36);(5,6,10,15)(7,16)(8,13,22,32)(9,14,23,11)(12,24)(17,31)(18,29)(19,28)(20,27)(21,26,25,30)(33,36)(34,35);(2,4)(5,7)(6,12)(8,20)(9,24)(10,19)(11,16)(13,31)(14,28)(15,29)(17,21)(18,23)(22,34)(25,33)(26,35)(27,30)(32,36);(5,8,10,22)(6,13,15,32)(7,17,19,33)(9,25,23,21)(11,26,14,30)(12,27,29,35)(16,31,28,36)(18,34,24,20)

Polycyclic presentation:

2^9|b^2=egh,d^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,e^b=ei,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi

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, 378653);