#474025

order	512
nilpotency class	3
isoclinism	isoclinic class of #453995
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2812
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16893
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

887 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)(4,20)(5,11,28,39)(6,13)(7,12)(8,14)(9,18)(10,23)(16,43)(17,47,50,42)(19,40)(21,36,53,59)(22,35)(24,34)(25,33,52,60)(26,38,27,37)(29,32)(30,46)(31,51,58,49)(41,45)(44,56,64,63)(48,62,55,57)(54,61);(1,3,7,18,8,19,6,29)(2,9,12,40,14,32,13,15)(4,16,23,45,24,46,22,54)(5,17,26,51,28,50,27,49)(10,30,34,61,35,43,20,41)(11,31,38,47,39,58,37,42)(21,44,25,55,53,64,52,48)(33,56,59,57,60,63,36,62);(1,4,7,23,8,24,6,22)(2,10,13,20,14,35,12,34)(3,16,18,45,19,46,29,54)(5,25,27,21,28,52,26,53)(9,30,15,41,32,43,40,61)(11,36,38,33,39,59,37,60)(17,48,49,64,50,55,51,44)(31,57,47,63,58,62,42,56);(1,5)(2,11)(3,17)(4,21)(6,27)(7,26)(8,28)(9,31)(10,33)(12,38)(13,37)(14,39)(15,42)(16,44)(18,51)(19,50)(20,36)(22,52)(23,25)(24,53)(29,49)(30,56)(32,58)(34,59)(35,60)(40,47)(41,62)(43,63)(45,55)(46,64)(48,54)(57,61)

Polycyclic presentation:

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

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, 381221);