#298523

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59654	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1238 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)(6,10)(7,8)(11,12)(13,14)(15,23)(16,26)(17,21)(18,24)(19,20)(22,28)(25,27);(1,2,4,3)(6,11)(7,8)(10,12)(13,15,17,20)(14,19,21,23)(16,24,27,22)(18,26,28,25);(5,6,7,10,9,12,8,11)(13,16,18,19,17,27,28,23)(14,20,22,26,21,15,24,25);(1,3,4,2)(6,12)(10,11)(14,21)(16,27)(19,23)(22,24)

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,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, 205719);