#146322

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1726
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,10,17,12,19,32,48,13,18,11,35,9,16,31,47)(7,20,23,50,24,14,36,38,25,41,33,51,22,39,57,40)(8,26,28,59,29,45,61,46,30,43,34,60,27,15,42,44)(21,37,53,63,55,58,66,68,56,64,54,65,52,49,62,67);(2,4)(5,7)(6,14,18,39)(8,21)(9,22)(10,33)(11,23)(12,24)(13,25)(15,37,45,64)(16,20,19,41)(17,40,35,38)(26,58,43,49)(27,52)(28,54)(29,55)(30,56)(31,36)(32,57)(34,53)(42,66)(44,65,46,63)(47,51,48,50)(59,67,60,68)(61,62);(5,8,11,28,12,29,31,61,13,30,10,34,9,27,32,42)(6,15,17,46,19,26,48,60,18,45,35,44,16,43,47,59)(7,21,23,54,24,55,36,62,25,56,33,53,22,52,57,66)(14,37,40,63,41,58,50,68,39,64,38,65,20,49,51,67)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ch,f^a=fi,g^a=gi,e^b=ei,e^c=ei

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, 53518);