#271310

order	512
nilpotency class	3
isoclinism	isoclinic class of #271259
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

981 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 178506);