#357094

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	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=efg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,f^c=fi,g^c=gi

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, 264290);