#268784

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

993 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:

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

Polycyclic presentation:

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

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, 175980);