#346657

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
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	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253853);