#213259

order	512
nilpotency class	3
isoclinism	isoclinic class of #19713
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	#3169	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,d^c=dh

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, 120455);