#379043

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	#23438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1320 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)(5,6,11,16,12,17,10,27)(7,13,9,15,20,29,19,34)(8,21,25,33,26,32,24,14)(18,31,23,36,35,30,22,28);(1,2,4,3)(5,7,11,9,12,20,10,19)(6,13,16,15,17,29,27,34)(8,22,24,35,26,23,25,18)(14,30,32,36,33,31,21,28);(5,8,10,24,12,26,11,25)(6,14,16,21,17,33,27,32)(7,18,19,22,20,35,9,23)(13,28,15,31,29,36,34,30);(1,3,4,2)(5,9,11,20,12,19,10,7)(6,15,16,29,17,34,27,13)(8,23,25,35,26,22,24,18)(14,31,21,36,33,30,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=f,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,d^b=di,g^c=gi

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, 286239);