#359897

order	512
nilpotency class	3
isoclinism	isoclinic class of #190432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267093);