#380717

order	512
nilpotency class	3
isoclinism	isoclinic class of #29748
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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287913);