#373678

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14127
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2109 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317339, #72317352
maximal normal subgroup of	#72317339, #72317352

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280874);