#282514

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,9,15,11,16,10,14)(7,12,17,20,18,13,8,19)(21,22,27,33,26,24,25,29)(23,28,34,36,30,32,31,35);(2,4)(5,7,9,17,11,18,10,8)(6,12,15,20,16,13,14,19)(21,23,27,34,26,30,25,31)(22,28,33,36,24,32,29,35);(5,8,11,17)(6,13,16,12)(7,9,18,10)(14,19,15,20)(21,24,26,22)(23,28,30,32)(25,29,27,33)(31,36,34,35);(2,4)(5,9,11,10)(6,14,16,15)(7,8,18,17)(12,20,13,19)(22,24)(23,30)(29,33)(31,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=h,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, 189710);