#394021

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

3914 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)(9,24)(10,14)(11,15)(12,18)(17,33)(19,27)(20,32)(21,26)(22,30)(23,31)(25,29)(28,34)(35,36);(5,7)(6,12)(8,21)(9,19)(10,18)(11,20)(13,28)(14,32)(15,27)(16,24)(17,22)(23,35)(25,34)(26,30)(29,36)(31,33);(2,4)(5,8,11,23)(6,13,15,31)(7,17,20,34)(9,22,10,25)(12,26,27,36)(14,29,24,30)(16,33,32,28)(18,21,19,35);(5,6)(7,12)(8,13)(9,24)(10,14)(11,15)(16,19)(17,26)(18,32)(20,27)(21,28)(22,30)(23,31)(25,29)(33,35)(34,36)

Polycyclic presentation:

2^9|c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,h^b=hi,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, 301217);