#376197

order	512
nilpotency class	3
isoclinism	isoclinic class of #219491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1503 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)(5,6,11,15,12,17,10,16)(7,18,21,29,8,22,20,13)(9,24,26,31,27,30,25,14)(19,33,35,36,23,28,34,32);(1,2,4,3)(5,7,10,20,12,8,11,21)(6,13,15,18,17,29,16,22)(9,19,25,34,27,23,26,35)(14,28,24,32,31,33,30,36);(5,8,12,7)(6,13,17,29)(9,23,27,19)(10,20,11,21)(14,28,31,33)(15,22,16,18)(24,36,30,32)(25,34,26,35);(1,3,4,2)(5,9)(6,14,17,31)(7,19)(8,23)(10,25)(11,26)(12,27)(13,28,29,33)(15,24,16,30)(18,32,22,36)(20,34)(21,35)

Polycyclic presentation:

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

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, 283393);