#146402

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#386
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10714
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

875 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341758
maximal normal subgroup of	#72341758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53598);