#282302

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
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	#25599	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

845 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)(5,6,9,11,10,7,8,12)(13,14,16,21,17,20,24,19)(15,18,23,25,22,26,28,27);(2,4)(5,7,9,12,10,6,8,11)(13,15,16,23,17,22,24,28)(14,18,21,25,20,26,19,27);(5,6)(7,10)(8,11)(9,12)(14,19)(15,22)(16,24)(18,25)(20,21)(26,27);(2,4)(5,8,10,9)(6,11,7,12)(14,20)(15,22)(19,21)(23,28)

Polycyclic presentation:

2^9|a^2=f,b^2=f,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,f^c=fh,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, 189498);