#355590

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	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1100 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efg,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=ce,f^c=fi,g^c=gi

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, 262786);