#348281

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

829 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,c^b=ch,f^c=fh,g^c=gi,f^d=fi,g^d=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, 255477);