#147810

order	512
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10995
prime radical	#60749	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1034 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,e^b=ehi,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,e^d=ei,g^d=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, 55006);