#148736

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1804
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=eh,f^c=fi,h^c=hi

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, 55932);