#380721

order	512
nilpotency class	3
isoclinism	isoclinic class of #29751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1248 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)(10,12)(13,14,17,23,18,22,16,21)(15,19,24,27,26,20,25,28);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15,16,25,18,26,17,24)(14,19,21,28,22,20,23,27);(2,4)(7,11)(10,12)(13,15)(14,20)(16,24)(17,25)(18,26)(19,22)(21,28)(23,27);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,21,22,23)(15,24,26,25)(19,27,20,28)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287917);