#355848

order	512
nilpotency class	3
isoclinism	isoclinic class of #3192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×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×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1254 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)(6,9)(7,11)(13,14)(15,20)(16,25)(17,26)(18,22)(19,23)(21,27)(24,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,24,19,25,17,21)(14,20,22,28,23,16,26,27);(6,9)(10,12)(13,16)(14,21)(15,22)(17,28)(18,27)(19,20)(23,24)(25,26);(1,3,4,2)(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cei,f^c=fi

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, 263044);