#449559

order	512
nilpotency class	3
isoclinism	isoclinic class of #417454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35659	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2118 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=i,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,h^c=hi,f^d=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, 356755);