#364935

order	512
nilpotency class	3
isoclinism	isoclinic class of #196161
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6)(7,11)(8,18,21,30)(9,13)(10,14)(12,20,28,23)(15,32,19,35)(16,26)(17,27)(22,29)(24,31)(25,33,36,34);(1,2,4,3)(5,7,10,17)(6,11,14,27)(8,19)(9,22,24,16)(12,25)(13,29,31,26)(15,21)(18,35)(20,33)(23,34)(28,36)(30,32);(5,8)(6,12)(7,15)(9,23)(10,21)(11,25)(13,30)(14,28)(16,34)(17,19)(18,31)(20,24)(22,33)(26,35)(27,36)(29,32);(1,3,4,2)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272131);