#363906

order	512
nilpotency class	3
isoclinism	isoclinic class of #363905
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13956
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2457 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)(3,13,17,38)(4,9)(5,21,23,51)(6,11)(7,12)(8,16,28,46)(10,22,33,53)(14,36,41,55)(15,42)(18,47,49,50)(19,31)(20,32)(24,34)(25,35)(26,45,29,40)(27,44)(30,48,57,52)(37,59)(39,54)(43,60)(56,62)(58,63)(61,64);(1,3)(2,8,12,28)(4,14)(5,15)(6,16)(7,17)(9,29,32,26)(10,27,33,42)(11,13,35,38)(18,39)(19,40)(20,41)(21,37,51,60)(22,43)(23,44)(24,45)(25,46)(30,56,57,54)(31,36,34,55)(47,58,50,64)(48,61)(49,62)(52,63)(53,59);(1,4,7,20)(2,9,12,32)(3,14,17,41)(5,18,23,49)(6,24,25,19)(8,26,28,29)(10,30,33,57)(11,34,35,31)(13,36,38,55)(15,39,44,62)(16,45,46,40)(21,50,51,47)(22,52,53,48)(27,54,42,56)(37,58,60,64)(43,63,59,61);(1,5)(2,10,12,33)(3,15)(4,18)(6,22)(7,23)(8,27,28,42)(9,30,32,57)(11,21,35,51)(13,37,38,60)(14,39)(16,43)(17,44)(19,48)(20,49)(24,52)(25,53)(26,54,29,56)(31,47,34,50)(36,58,55,64)(40,61)(41,62)(45,63)(46,59)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251694, #72251709, #72251731
maximal normal subgroup of	#72251694, #72251709, #72251731
maximal quotient of	#72255906

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271102);