#148183

order	512
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#11069
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1334 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72267920

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55379);