#376149

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1536 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283345);