#402131

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

1080 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)(5,6,9,10)(7,11,8,12)(13,14,16,18)(15,17,19,20)(21,22)(23,32)(24,30)(25,29)(26,31)(27,33)(28,34)(35,36);(6,10)(11,12)(13,15)(14,17)(16,19)(18,20)(21,23,25,34,26,35,24,33)(22,27,30,36,31,28,29,32);(5,7,9,8)(6,11,10,12)(15,19)(17,20)(21,23,25,34,26,35,24,33)(22,28,30,32,31,27,29,36);(2,4)(5,8,9,7)(6,12,10,11)(21,23,24,33,26,35,25,34)(22,28,29,36,31,27,30,32)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=ei,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=di

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, 309327);