#282538

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24758	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

783 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)(5,6,9,7,10,12,8,11)(13,14,16,20,17,19,25,24)(15,18,23,26,22,21,27,28);(2,4)(5,7,8,6,10,11,9,12)(13,15,16,23,17,22,25,27)(14,18,20,26,19,21,24,28);(5,6,10,12)(7,9,11,8)(13,14)(15,21)(16,24)(17,19)(18,22)(20,25)(23,26)(27,28);(2,4)(5,8,10,9)(6,7,12,11)(14,19)(15,22)(20,24)(23,27)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=hi,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189734);