#147672

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1874
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

1004 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)(5,6,10,16,12,17,11,34)(7,18,22,47,23,48,21,44)(8,14,27,42,28,43,26,41)(9,29,31,45,33,61,32,15)(13,30,37,52,38,51,36,20)(19,46,24,54,50,67,53,66)(25,56,59,65,60,64,58,40)(35,55,39,57,63,68,62,49);(5,7)(6,13)(8,24)(9,30)(10,22)(11,21)(12,23)(14,39)(15,44)(16,37)(17,38)(18,29)(19,26)(20,32)(25,57)(27,50)(28,53)(31,52)(33,51)(34,36)(35,41)(40,46)(42,63)(43,62)(45,47)(48,61)(49,60)(54,56)(55,58)(59,68)(64,66)(65,67);(2,4)(5,8)(6,14)(7,19)(9,25)(10,27)(11,26)(12,28)(13,35)(15,40)(16,42)(17,43)(18,46)(20,49)(21,53)(22,24)(23,50)(29,56)(30,55)(31,59)(32,58)(33,60)(34,41)(36,62)(37,39)(38,63)(44,66)(45,65)(47,54)(48,67)(51,68)(52,57)(61,64)

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,h^a=hi,c^b=cf,d^b=dh,f^b=fi,h^b=hi,f^c=fi

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, 54868);