#495681

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3002	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

941 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402877);