#366316

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 273512);