#277066

order	512
nilpotency class	3
isoclinism	isoclinic class of #267131
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

634 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)(5,6,10,18,9,13,8,12)(7,14,15,11,17,19,16,20)(21,22,24,28,26,30,25,29)(23,27,31,35,33,34,32,36);(1,2,4,3)(21,23,26,33)(22,27,30,34)(24,31,25,32)(28,35,29,36);(5,7,9,17)(6,11,13,20)(8,15,10,16)(12,19,18,14)(22,28)(23,31)(24,25)(27,34)(29,30)(32,33);(1,3,4,2)(5,8,9,10)(6,12,13,18)(7,15,17,16)(11,19,20,14)(21,24,26,25)(22,28,30,29)(23,31,33,32)(27,35,34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 184262);