#213356

order	512
nilpotency class	3
isoclinism	isoclinic class of #19733
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	#19619	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

677 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)(4,12)(6,13)(8,14)(10,15)(11,16)(17,18,21,27,23,28,22,26)(19,29,20,24,31,32,30,25);(1,2)(3,10)(4,8)(5,7)(6,11)(9,15)(12,14)(13,16)(17,19,21,20,23,31,22,30)(18,24,26,29,28,25,27,32);(1,3,4,9,5,11,12,16)(2,6,8,13,7,10,14,15)(17,19,22,30,23,31,21,20)(18,25,27,29,28,24,26,32);(2,7)(3,11)(8,14)(9,16)(17,20,21,31,23,30,22,19)(18,24,27,32,28,25,26,29)

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,c^b=ci,d^b=di,d^c=dhi

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, 120552);