#358859

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2612 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)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(2,4)(10,13)(14,16)(17,19,21,25)(18,23,24,30)(20,26,27,31)(22,29,28,32);(9,11,12,15)(10,14,13,16)(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=g,e^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266055);