#496346

order	512
nilpotency class	4
isoclinism	isoclinic class of #496344
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, prime radical, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

728 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403542);