#372873

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,18)(17,20)(21,22,25,30,24,29,26,28)(23,31,32,36,33,27,34,35);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,16)(14,18,20,19)(21,23,26,34,24,33,25,32)(22,27,28,36,29,31,30,35);(2,4)(5,7,9,8)(6,11,12,10)(13,16,17,15)(14,19,20,18)(22,28)(23,32)(25,26)(29,30)(33,34)(35,36);(5,8,9,7)(6,11,12,10)(14,20)(18,19)(23,33)(27,31)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ei,d^2=efgi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280069);