#148180

order	512
nilpotency class	4
isoclinism	isoclinic class of #13350
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

1330 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 1330 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)(3,15,19,33,20,41,17,9)(4,21,25,53,26,55,23,54)(6,12,29,38,30,40,27,43)(10,34,36,59,37,58,42,24)(16,46,48,63,49,52,22,56)(18,45,51,39,28,60,50,32)(31,47,44,64,62,61,35,57);(1,3,5,17,8,20,7,19)(2,9,11,15,14,33,13,41)(4,22,23,49,26,48,25,16)(6,28,27,51,30,18,29,50)(10,35,36,31,37,44,42,62)(12,39,38,60,40,32,43,45)(21,52,53,56,55,46,54,63)(24,57,58,61,59,64,34,47);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,36)(12,21)(13,42)(14,37)(15,44)(17,22)(18,47)(19,48)(20,49)(27,58)(28,61)(29,34)(30,59)(32,56)(33,62)(35,41)(38,53)(39,63)(40,55)(43,54)(45,46)(50,57)(51,64)(52,60)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^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,f^c=fi,h^c=hi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341811, #72341819
maximal normal subgroup of	#72341811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55376);