#449827

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1479 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)(10,11)(13,14,18,19,17,22,15,23)(16,26,25,20,27,28,24,21);(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,15,17,18)(14,19,22,23)(16,24,27,25)(20,28,21,26);(2,4)(13,16)(14,20)(15,24)(17,27)(18,25)(19,28)(21,22)(23,26);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,25)(17,27)(18,24)(19,28)(20,22)(23,26)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^c=fi,h^d=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, 357023);