#498380

order	512
nilpotency class	4
isoclinism	isoclinic class of #498379
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23029
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

587 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285481, #72285570, #72285705
maximal normal subgroup of	#72285481, #72285570, #72285705
maximal quotient of	#72287271, #72287490, #72309413, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405576);