#215234

order	512
nilpotency class	3
isoclinism	isoclinic class of #163012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122430);