#366752

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1708 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)(3,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(19,27)(23,31)(28,32)(29,30);(1,2)(3,4)(10,13)(14,16)(17,18,20,24,21,25,22,26)(19,28,30,23,27,32,29,31);(2,4)(6,8)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,29)(21,27)(22,30)(24,28)(25,31)(26,32);(10,13)(14,16)(17,20,21,22)(18,24,25,26)(19,29,27,30)(23,28,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273948);