#395790

order	512
nilpotency class	3
isoclinism	isoclinic class of #395789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14727
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1733 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)(3,14,18,38)(4,10)(5,25,27,11)(6,12)(7,29)(8,13)(9,17,32,19)(15,40,43,33)(16,44)(20,48,42,30)(21,34,51,55)(22,35)(23,53)(24,36)(26,56,28,37)(31,45)(39,47)(41,60)(46,54)(49,63)(50,59,52,64)(57,61)(58,62);(1,3,8,18)(2,9,13,32)(4,20,24,42)(5,16,27,47)(6,19,7,17)(10,33,36,40)(11,31,25,54)(12,38,29,14)(15,22,43,23)(21,49,51,61)(26,46,28,45)(30,53,48,35)(34,58,55,60)(37,44,56,39)(41,50,62,52)(57,64,63,59);(1,4,6,22,8,24,7,23)(2,10,12,35,13,36,29,53)(3,15,17,42,18,43,19,20)(5,21,26,50,27,51,28,52)(9,30,14,33,32,48,38,40)(11,34,37,59,25,55,56,64)(16,41,45,61,47,62,46,49)(31,57,39,58,54,63,44,60);(1,5)(2,11)(3,16)(4,21)(6,26)(7,28)(8,27)(9,31)(10,34)(12,37)(13,25)(14,39)(15,41)(17,45)(18,47)(19,46)(20,49)(22,50)(23,52)(24,51)(29,56)(30,57)(32,54)(33,58)(35,59)(36,55)(38,44)(40,60)(42,61)(43,62)(48,63)(53,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318497, #72318509, #72318571
maximal normal subgroup of	#72318497, #72318509, #72318571

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302986);