#498581

order	512
nilpotency class	4
isoclinism	isoclinic class of #498581
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23188
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33719	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

643 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=e,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,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285481, #72285570, #72285959
maximal normal subgroup of	#72285481, #72285570, #72285959
maximal quotient of	#72287346, #72287521, #72287705, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405777);