#148318

order	512
nilpotency class	4
isoclinism	isoclinic class of #134691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,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, 55514);