#147637

order	512
nilpotency class	4
isoclinism	isoclinic class of #137659
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1014 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=dh,h^b=hi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281903, #72281975, #72282210
maximal normal subgroup of	#72281903, #72281975, #72282210

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54833);