#364930

order	512
nilpotency class	3
isoclinism	isoclinic class of #196290
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

1680 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)(7,12)(8,19,23,32)(9,14)(10,25)(11,15)(13,21,31,22)(16,33,34,30)(17,27)(18,29)(20,36,35,26)(24,28);(1,2,4,3)(5,7,11,18)(6,12,15,29)(8,20)(9,24,10,17)(13,30)(14,28,25,27)(16,21)(19,36)(22,34)(23,35)(26,32)(31,33);(5,8)(6,13)(7,16)(9,22)(10,21)(11,23)(12,26)(14,32)(15,31)(17,35)(18,34)(19,25)(20,24)(27,30)(28,33)(29,36);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,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, 272126);