#120362

order	512
nilpotency class	3
isoclinism	isoclinic class of #120361
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

431 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fhi,e^c=eh,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27558);