#147648

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1032 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)(8,22,24,13)(9,26)(10,32)(11,15)(12,19)(14,30)(16,29)(18,43,46,64)(20,44)(21,50)(23,53,28,61)(25,38,55,52)(27,42)(31,58)(33,39)(34,45,51,48)(35,60)(36,49)(37,56,40,62)(41,63)(47,66,57,68)(54,65,59,67);(1,2)(3,4)(5,7)(6,12)(8,18)(9,27)(10,33)(11,20)(13,34)(14,39)(15,36)(16,42)(17,26)(19,30)(21,29)(22,51)(23,54)(24,46)(25,57)(28,59)(31,60)(32,35)(37,66)(38,67)(40,68)(41,50)(43,53)(44,58)(45,56)(47,55)(48,62)(49,63)(52,65)(61,64);(2,4)(5,8)(6,13)(7,18)(9,28)(10,25)(11,24)(12,34)(14,40)(15,22)(16,38)(17,43)(19,48)(20,46)(21,47)(23,31)(26,53)(27,59)(29,55)(30,62)(32,52)(33,57)(35,65)(36,51)(37,41)(39,68)(42,67)(44,64)(45,49)(50,66)(54,60)(56,63)(58,61)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi,d^c=di,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, 54844);