#495768

order	512
nilpotency class	4
isoclinism	isoclinic class of #33568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₄×B(4,𝔽₂)
prime radical	#33564	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1003 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250212, #72317420
maximal normal subgroup of	#72250212, #72317420
maximal quotient of	#72257236

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402964);