#299122

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
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

1031 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:

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

Polycyclic presentation:

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

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, 206318);