#394077

order	512
nilpotency class	3
isoclinism	isoclinic class of #394020
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3982 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)(9,21)(12,26)(13,22)(15,28)(18,23)(20,32)(24,35)(30,34);(5,6)(7,15)(8,12)(9,22)(10,14)(11,18)(13,21)(16,29)(17,28)(19,26)(20,33)(23,25)(24,30)(27,32)(31,36)(34,35);(2,4)(5,7,10,17)(6,11,14,25)(8,16,19,31)(9,23,21,18)(12,24,26,35)(13,28,22,15)(20,34,32,30)(27,36,33,29);(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,27)(14,26)(15,29)(17,31)(18,30)(21,32)(22,33)(23,34)(25,35)(28,36)

Polycyclic presentation:

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

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, 301273);