#361365

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1368 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ei,d^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg

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, 268561);