#417192

order	512
nilpotency class	3
isoclinism	isoclinic class of #394253
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1580 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)(7,11)(8,9)(10,12)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,28)(26,27);(1,2,3,4)(6,9)(10,11)(14,18)(15,22,23,26)(19,27,21,28)(20,25);(13,15,17,26,16,23,24,22)(14,19,20,27,18,21,25,28);(5,7,8,12)(6,10,9,11)(14,18)(19,21)(20,25)(27,28)

Polycyclic presentation:

2^9|b^2=eh,c^2=fg,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi

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, 324388);