#364970

order	512
nilpotency class	3
isoclinism	isoclinic class of #196134
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

1660 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)(8,18,21,12)(9,22)(10,13)(14,24)(15,25,31,36)(16,27)(17,33)(19,35,34,28)(20,29,23,30)(26,32);(1,2,4,3)(5,7,10,16)(6,11,13,27)(8,19)(9,17,24,32)(12,28)(14,26,22,33)(15,23)(18,35)(20,31)(21,34)(25,30)(29,36);(5,8)(6,12)(7,15)(9,23)(10,21)(11,25)(13,18)(14,30)(16,31)(17,34)(19,32)(20,24)(22,29)(26,35)(27,36)(28,33);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=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, 272166);