#149002

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

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	#1835
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

434 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)(7,19)(8,26,29,57)(9,31)(10,15)(11,34)(12,18)(13,22)(14,28,40,60)(16,20)(17,32)(21,49,52,35)(23,44)(24,36)(25,46)(27,58,48,41)(30,61,59,39)(33,42)(37,55)(38,47)(43,51,64,68)(45,54)(50,56,66,65)(53,62,67,63);(1,2)(3,4)(5,7,20,47,32,54,10,23,12,25,9,22,11,24,33,55)(6,13,15,19,34,44,16,36,18,38,42,46,17,37,31,45)(8,21,48,66,59,67,28,51,29,52,27,50,30,53,60,68)(14,35,39,43,58,63,26,56,40,49,61,64,41,62,57,65);(2,4)(5,8,9,27,32,59,33,60,12,29,20,48,11,30,10,28)(6,14,15,39,34,58,16,26,18,40,42,61,17,41,31,57)(7,21,22,50,54,67,55,68,25,52,47,66,24,53,23,51)(13,35,19,43,44,63,36,56,38,49,46,64,37,62,45,65)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,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, 56198);