#420649

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1231 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:

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

Polycyclic presentation:

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

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, 327845);