#354376

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261572);