#148317

order	512
nilpotency class	4
isoclinism	isoclinic class of #134691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

666 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)(7,17)(8,23,25,13)(9,27)(10,30)(11,15)(12,20)(14,18)(16,28)(19,44,46,37)(21,50)(22,39)(24,54,56,57)(26,35,55,53)(29,58)(31,45,51,67)(32,49)(33,60)(34,42,52,64)(36,43)(38,41)(40,48)(47,62,66,65)(59,68,61,63);(1,2)(3,4)(5,7,18,41,10,21,29,49,11,22,43,60,28,48,9,20)(6,12,27,40,16,33,36,39,15,32,58,50,30,38,14,17)(8,19,42,63,26,47,56,67,25,46,64,68,55,66,24,45)(13,31,57,62,35,59,52,44,23,51,54,65,53,61,34,37);(2,4)(5,8)(6,13)(7,19)(9,24)(10,26)(11,25)(12,31)(14,34)(15,23)(16,35)(17,37)(18,42)(20,45)(21,47)(22,46)(27,57)(28,55)(29,56)(30,53)(32,51)(33,59)(36,52)(38,61)(39,44)(40,62)(41,63)(43,64)(48,66)(49,67)(50,65)(54,58)(60,68)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,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, 55513);