#271018

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
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	#36493	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1113 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)(7,12)(8,16)(9,11)(10,14)(13,15)(17,19)(18,20)(21,22,25,29,26,30,24,28)(23,31,32,35,33,36,34,27);(1,2)(3,4)(6,11)(7,9)(8,17)(12,14)(13,19)(16,20);(2,4)(5,7,10,9)(6,12,14,11)(8,15,17,18)(13,19,20,16)(21,23,26,33)(22,27,30,35)(24,32,25,34)(28,31,29,36);(5,8,10,17)(6,13,14,20)(7,15,9,18)(11,16,12,19)(21,24,26,25)(22,28,30,29)(23,32,33,34)(27,31,35,36)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,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=dh,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, 178214);