#268216

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1132 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fh,e^d=ei,f^d=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175412);