#318503

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

684 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:

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

Polycyclic presentation:

2^9|b^2=egh,c^2=i,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dhi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225699);