#148829

order	512
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11222
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 622 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)(3,15,19,33,20,42,17,9)(4,21,23,54,26,55,25,53)(6,12,29,39,30,41,27,43)(10,34,37,24,38,58,36,59)(16,46,22,56,48,52,51,63)(18,45,50,40,28,60,49,32)(31,47,35,57,62,61,44,64);(1,3,5,17,8,20,7,19)(2,9,11,15,14,33,13,42)(4,22,23,48,26,51,25,16)(6,28,27,50,30,18,29,49)(10,35,36,31,38,44,37,62)(12,40,39,60,41,32,43,45)(21,52,53,56,55,46,54,63)(24,57,58,61,59,64,34,47);(1,4,5,23,8,26,7,25)(2,10,11,36,14,38,13,37)(3,16,17,22,20,48,19,51)(6,24,27,58,30,59,29,34)(9,31,15,44,33,62,42,35)(12,21,39,53,41,55,43,54)(18,47,49,57,28,61,50,64)(32,56,45,46,40,63,60,52)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^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,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 subgroup of	#72341811
maximal normal subgroup of	#72341811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56025);