#149219

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	#1846
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

478 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=eh,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,g^b=gi,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, 56415);