#356896

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=ei,g^b=gi,d^c=di

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, 264092);