#496349

order	512
nilpotency class	4
isoclinism	isoclinic class of #496348
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21794
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#535	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,d^b=di,e^c=ei,g^c=gh,f^d=fi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403545);