#147683

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10969
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54879);