#394079

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
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3886 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)(5,6,11,15)(7,12,19,31)(8,9,13,22)(10,24,27,14)(16,28,34,36)(17,20,35,29)(18,32,25,30)(21,33,26,23);(5,7)(6,12)(8,20)(9,17)(10,25)(11,19)(13,29)(14,32)(15,31)(16,21)(18,27)(22,35)(23,28)(24,30)(26,34)(33,36);(2,4)(5,8,11,13)(6,9,15,22)(7,16,19,34)(10,26,27,21)(12,28,31,36)(14,33,24,23)(17,30,35,32)(18,29,25,20);(5,9)(6,13)(7,17)(8,15)(10,23)(11,22)(12,29)(14,26)(16,30)(18,36)(19,35)(20,31)(21,24)(25,28)(27,33)(32,34)

Polycyclic presentation:

2^9|a^2=i,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, 301275);