#148738

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2101
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,e^c=eh,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, 55934);