#271657

order	512
nilpotency class	3
isoclinism	isoclinic class of #271643
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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1036 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)(7,11)(8,10)(9,13)(12,16)(14,17)(15,19)(18,20)(21,22,26,29,27,30,25,34)(23,28,24,32,31,36,33,35);(1,2)(3,4)(6,10)(8,16)(11,17)(12,13)(14,20)(18,19)(21,23)(22,28)(24,26)(25,33)(27,31)(29,32)(30,36)(34,35);(2,4)(5,7,9,15)(6,11,13,19)(8,14,16,20)(10,17,12,18)(21,24,27,33)(22,28,30,36)(23,25,31,26)(29,35,34,32);(6,10)(8,16)(11,17)(12,13)(14,20)(18,19)(21,25,27,26)(22,29,30,34)(23,24,31,33)(28,35,36,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 178853);