#493952

order	512
nilpotency class	3
isoclinism	isoclinic class of #188761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1082 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=ci,d^b=di,g^b=gi,d^c=di,g^d=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401148);