#411714

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1906 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)(6,12)(8,22)(9,10)(13,26)(14,29)(15,16)(17,28)(18,32)(19,20)(23,24)(31,35)(33,34);(2,4)(5,6)(7,17)(8,14)(9,16)(10,12)(11,15)(13,19)(18,31)(20,28)(21,26)(22,25)(23,30)(24,29)(27,33)(32,36)(34,35);(5,7)(6,13)(8,18)(9,19)(10,20)(11,21)(12,17)(14,27)(15,28)(16,26)(22,32)(23,33)(24,34)(25,31)(29,36)(30,35);(5,8,9,23,11,24,10,22)(6,14,16,30,15,29,12,25)(7,18,19,33,21,34,20,32)(13,27,26,35,28,36,17,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318910);