#239891

order	512
nilpotency class	3
isoclinism	isoclinic class of #233958
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35166	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1390 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)(8,11)(10,12)(13,14)(15,23)(16,21)(17,22)(18,20)(19,25)(24,28)(26,27);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,26)(17,25)(18,24)(20,27)(21,28)(22,23);(1,3,4,2)(6,7)(9,10)(11,12)(14,20)(16,18)(19,27)(21,22)(23,28)(24,26);(13,16,17,18)(14,21,22,20)(15,24,25,26)(19,27,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147087);