#215230

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=h,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, 122426);