#239757

order	512
nilpotency class	3
isoclinism	isoclinic class of #233984
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1216 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)(7,12)(8,11)(9,10)(13,14)(15,19)(16,20)(17,26)(18,21)(22,25)(23,27)(24,28);(1,2)(3,4)(5,7,8,9)(6,10,11,12)(13,15)(14,19)(16,25)(17,23)(18,24)(20,22)(21,28)(26,27);(1,3,4,2)(6,10)(7,9)(11,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,23)(24,27)(25,28);(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,28,22)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146953);