#148293

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#2077
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

650 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,f^b=fi,g^b=gi,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, 55489);