#355852

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1171 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263048);