#282045

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

869 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)(5,6,9,12,8,11,10,7)(13,14,17,20,18,21,16,25)(15,19,23,26,24,27,22,28);(2,4)(5,7,9,6,8,12,10,11)(13,15,17,23,18,24,16,22)(14,19,20,26,21,27,25,28);(6,7)(9,10)(11,12)(13,14)(15,19)(16,20)(17,25)(18,21)(22,26)(23,28)(24,27);(2,4)(13,16,18,17)(14,20,21,25)(15,22,24,23)(19,26,27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,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, 189241);