#148826

order	512
nilpotency class	4
isoclinism	isoclinic class of #13432
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

634 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,e^b=eh,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	#72342095
maximal normal subgroup of	#72342095

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56022);