#366363

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1872 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273559);