#453920

order	512
nilpotency class	3
isoclinism	isoclinic class of #453920
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16224
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

626 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)(3,15,19,9)(4,20)(5,26,28,11)(6,14)(7,13)(8,12)(10,23)(16,44,46,35)(17,33)(18,34,30,40)(21,42,51,32)(22,49,53,62)(24,37)(25,38)(27,43)(29,39,31,41)(36,54,56,55)(45,64)(47,60)(48,59)(50,63)(52,58)(57,61);(1,3,7,18,8,19,6,30)(2,9,13,40,12,15,14,34)(4,21,23,16,25,51,24,46)(5,27,29,47,28,17,31,48)(10,35,38,42,37,44,20,32)(11,33,41,59,26,43,39,60)(22,52,55,57,53,50,54,45)(36,61,62,63,56,64,49,58);(1,4)(2,10,12,37)(3,16,19,46)(5,22)(6,23)(7,24)(8,25)(9,32)(11,36,26,56)(13,20,14,38)(15,42)(17,45,27,57)(18,21,30,51)(28,53)(29,54)(31,55)(33,58)(34,35)(39,62,41,49)(40,44)(43,63)(47,52,48,50)(59,64)(60,61);(1,5,6,31,8,28,7,29)(2,11,14,39,12,26,13,41)(3,17,18,48,19,27,30,47)(4,22,23,55,25,53,24,54)(9,33,40,59,15,43,34,60)(10,36,38,62,37,56,20,49)(16,45,21,50,46,57,51,52)(32,58,44,64,42,63,35,61)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361116);