#147646

order	512
nilpotency class	4
isoclinism	isoclinic class of #136628
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10969
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1023 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)(3,13)(4,18,20,9)(5,22)(6,27)(7,11)(8,15)(10,25)(12,24)(14,38,41,59)(16,39)(17,44)(19,47,50,53)(21,34,49,46)(23,37)(26,54)(28,35)(29,57,32,40)(30,43)(31,55)(33,51,45,63)(36,56)(42,61,52,62)(48,60,64,58);(1,3)(2,8)(4,14)(5,23)(6,28)(7,16)(9,32)(10,35)(11,30)(12,37)(13,22)(15,25)(17,24)(18,29)(19,48)(20,41)(21,52)(26,55)(27,31)(33,61)(34,58)(36,44)(38,47)(39,54)(40,51)(42,49)(43,56)(45,62)(46,60)(50,64)(53,59)(57,63);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,29)(10,33)(11,18)(12,34)(13,38)(15,40)(16,41)(17,42)(22,53)(23,48)(24,49)(25,51)(26,50)(27,46)(28,52)(30,32)(31,58)(35,62)(36,45)(37,60)(39,59)(43,57)(44,61)(47,54)(55,64)(56,63)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280034, #72280256, #72280708, …
maximal normal subgroup of	#72280034, #72280256, #72280708, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54842);