#262174

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	#45999	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1056 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=h,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, 169370);