#149218

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=eh,d^2=h,e^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, 56414);