#368858

order	512
nilpotency class	3
isoclinism	isoclinic class of #368855
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14023
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1649 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=i,e^2=i,f^2=i,g^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,f^b=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264906, #72264921, #72264989
maximal normal subgroup of	#72264906, #72264921, #72264989
maximal quotient of	#72265254

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276054);