#271289

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

959 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)(7,16)(8,13)(9,14)(10,19)(11,15)(12,18)(17,20)(21,22,26,30,27,31,25,29)(23,28,24,33,32,35,34,36);(1,2)(3,4)(5,7,11,8)(6,12,15,20)(9,18,10,17)(13,19,16,14)(21,23)(22,28)(24,26)(25,34)(27,32)(29,36)(30,33)(31,35);(2,4)(5,8,11,7)(6,13,15,16)(9,18,10,17)(12,19,20,14)(21,24,27,34)(22,28,31,35)(23,25,32,26)(29,33,30,36);(5,9)(7,17)(8,18)(10,11)(12,20)(14,19)(21,25,27,26)(22,29,31,30)(23,24,32,34)(28,33,35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,d^2=i,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, 178485);