#282517

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
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	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 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)(5,6,8,12,9,7,10,11)(13,14,19,26,18,16,17,21)(15,22,23,28,24,20,25,27);(2,4)(5,7,8,11,9,6,10,12)(13,15,19,23,18,24,17,25)(14,20,26,27,16,22,21,28);(6,11)(7,12)(8,10)(13,16,18,14)(15,20,24,22)(17,21,19,26)(23,28,25,27);(2,4)(6,7)(11,12)(13,17,18,19)(14,21,16,26)(15,23,24,25)(20,27,22,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=i,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, 189713);