#262182

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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
prime radical	C₂×#1656	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1002 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)(5,6,7,9)(8,10,12,11)(13,14)(15,23)(16,21)(17,20)(18,22)(19,25)(24,28)(26,27);(1,2,4,3)(6,9)(8,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,28)(21,23)(22,27);(6,10)(8,12)(9,11)(14,20)(15,24)(17,18)(21,22)(25,26)(27,28);(1,3,4,2)(6,9)(10,11)(14,21)(19,23)(20,22)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169378);