#495106

order	512
nilpotency class	3
isoclinism	isoclinic class of #23414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1955
prime radical	C₂×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ch,f^b=fi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265577, #72265589, #72265606
maximal normal subgroup of	#72265577, #72265589, #72265606

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402302);