#355546

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1582 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,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, 262742);