#374127

order	512
nilpotency class	3
isoclinism	isoclinic class of #374124
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14134
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2367 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341332, #72341337, #72341346, …
maximal normal subgroup of	#72341332, #72341337, #72341346, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281323);