#471886

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2812
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16808
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,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, 379082);