#412315

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1248 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)(5,6,11,17)(7,18,22,32)(8,23,26,34)(9,27,29,36)(10,16,30,15)(12,21,33,20)(13,25,35,24)(14,19,31,28);(2,4)(5,7)(6,12)(8,9)(10,21)(11,22)(13,27)(14,23)(15,32)(16,18)(17,33)(19,24)(20,30)(25,28)(26,29)(31,34)(35,36);(5,8)(6,13)(7,19)(9,21)(10,25)(11,26)(12,31)(14,33)(15,34)(16,23)(17,35)(18,27)(20,29)(22,28)(24,30)(32,36);(5,9,10,28,11,29,30,19)(6,14,15,27,17,31,16,36)(7,8,21,25,22,26,20,24)(12,23,32,13,33,34,18,35)

Polycyclic presentation:

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

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, 319511);