#471458

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1142 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)(5,6,10,14)(7,15,19,27)(8,12,23,32)(9,24,25,13)(11,18,28,17)(16,29,34,35)(20,26,33,36)(21,31,22,30);(5,7,10,19)(6,11,14,28)(8,20)(9,18,25,17)(12,29)(13,27,24,15)(16,21)(22,34)(23,33)(26,30)(31,36)(32,35);(2,4)(5,8)(6,12)(7,16)(9,21)(10,23)(11,26)(13,30)(14,32)(15,29)(17,20)(18,33)(19,34)(22,25)(24,31)(27,35)(28,36);(8,21)(9,25)(12,30)(13,24)(15,27)(16,20)(17,18)(22,23)(26,29)(31,32)(33,34)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=egh,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, 378654);