#449655

order	512
nilpotency class	3
isoclinism	isoclinic class of #417454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1614 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)(13,14,18,15)(16,23)(17,20,27,22)(19,26)(21,24)(25,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,21,25,19)(17,22,27,20)(23,24,28,26);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,21)(17,26)(18,25)(20,28)(22,23)(24,27);(6,9)(10,12)(17,27)(20,22)(23,28)(24,26)

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,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, 356851);