#147647

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

1015 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,g^2=i,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	#72280321, #72280557, #72280708, …
maximal normal subgroup of	#72280321, #72280557, #72280708, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54843);