#223060

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
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	C₂³×C₄	C₂⁴
prime radical	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1022 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130256);