#357662

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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

2834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 264858);