#147687

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2033
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1002 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)(5,6,10,18,12,16,11,17)(7,19,24,48,22,49,23,44)(8,25,28,42,29,14,35,43)(9,30,32,45,34,46,33,15)(13,31,37,53,39,54,38,21)(20,47,52,67,51,68,26,57)(27,59,61,56,62,63,64,41)(36,65,40,60,66,58,55,50);(5,7)(6,13)(8,26)(9,31)(10,24)(11,23)(12,22)(14,40)(15,44)(16,39)(17,38)(18,37)(19,30)(20,28)(21,33)(25,55)(27,60)(29,52)(32,53)(34,54)(35,51)(36,42)(41,68)(43,66)(45,48)(46,49)(47,56)(50,62)(57,59)(58,61)(63,67)(64,65);(2,4)(5,8)(6,14)(7,20)(9,27)(10,35)(11,28)(12,29)(13,36)(15,41)(16,25)(17,43)(18,42)(19,47)(21,50)(22,51)(23,52)(24,26)(30,63)(31,58)(32,64)(33,61)(34,62)(37,55)(38,40)(39,66)(44,67)(45,56)(46,59)(48,57)(49,68)(53,60)(54,65)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,f^2=i,g^2=i,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,e^b=ei,f^b=fi,h^b=hi,f^c=fi,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, 54883);