#399215

order	512
nilpotency class	3
isoclinism	isoclinic class of #394048
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

1481 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306411);