#278810

order	512
nilpotency class	3
isoclinism	isoclinic class of #268891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=hi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=gi

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, 186006);