#6379197

order	512
nilpotency class	3
isoclinism	isoclinic class of #61109
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×D₄²
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×D₄×C₂²⋊C₄
prime radical	#63090	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4501 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,e^b=ei,f^b=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312363, #72312375, #72312427
maximal normal subgroup of	#72312363, #72312375, #72312427

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286393);