#356988

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1772 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)(9,10,12,11)(14,19)(15,23)(16,18)(20,28)(21,22)(24,27);(1,2)(3,4)(13,14,18,22,17,21,16,19)(15,24,26,27,23,20,25,28);(6,8)(9,10,12,11)(13,15,17,23)(14,20,21,24)(16,25,18,26)(19,27,22,28);(2,4)(9,11,12,10)(14,19)(16,18)(20,27)(21,22)(24,28)(25,26)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=fgi,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=dei,e^d=ei,h^d=hi

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, 264184);