#427310

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1364 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)(7,16)(8,13)(9,26)(10,14)(11,15)(12,19)(17,29)(18,20)(21,28)(22,36)(23,32)(24,30)(25,31)(27,34)(33,35);(5,7)(6,12)(8,22)(9,21)(10,19)(11,20)(13,29)(14,18)(15,28)(16,26)(17,24)(23,35)(25,34)(27,30)(31,33)(32,36);(5,8,11,25)(6,13,15,31)(7,17,20,35)(9,24,10,23)(12,27,28,36)(14,32,26,30)(16,33,18,29)(19,22,21,34);(2,4)(5,6)(7,18,20,16)(8,13)(9,14)(10,26)(11,15)(12,21,28,19)(17,29,35,33)(22,27,34,36)(23,30)(24,32)(25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334506);