#455830

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16270
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1072 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)(4,19)(5,11)(6,28)(7,12)(8,13)(9,17)(10,22)(15,44)(16,48)(18,33)(20,54)(21,39)(23,37)(24,36)(25,41)(26,40)(27,42)(29,34)(30,38)(31,47)(32,50)(35,46)(43,52)(45,63)(49,59)(51,60)(53,56)(55,62)(57,58)(61,64);(1,3,6,17,8,18,7,29)(2,9,12,14,13,34,28,33)(4,20,22,15,23,46,30,47)(5,16,26,51,27,52,25,50)(10,35,19,44,38,54,37,31)(11,32,41,43,42,60,40,48)(21,49,55,45,56,64,24,57)(36,61,53,63,62,59,39,58);(1,4,6,22,8,23,7,30)(2,10,12,19,13,38,28,37)(3,15,17,46,18,47,29,20)(5,24,25,56,27,55,26,21)(9,31,14,35,34,44,33,54)(11,39,40,62,42,53,41,36)(16,49,50,57,52,64,51,45)(32,59,48,63,60,61,43,58);(1,5)(2,11)(3,16)(4,21)(6,26)(7,25)(8,27)(9,32)(10,36)(12,41)(13,42)(14,43)(15,45)(17,51)(18,52)(19,53)(20,49)(22,55)(23,56)(24,30)(28,40)(29,50)(31,58)(33,48)(34,60)(35,61)(37,39)(38,62)(44,63)(46,64)(47,57)(54,59)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363026);