#289473

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

729 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=h,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196669);