#461424

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	#16391
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1418 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)(5,25,29,11)(6,13)(7,12)(8,14)(9,18)(10,23)(16,43)(17,42,49,60)(19,40)(21,51,53,37)(22,36)(24,35)(26,56,52,34)(27,38,28,39)(30,33)(31,46)(32,48,55,50)(41,45)(44,63,64,58)(47,59,57,62)(54,61);(1,3)(2,9)(4,16)(5,17)(6,30)(7,18)(8,19)(10,31)(11,32)(12,40)(13,15)(14,33)(20,41)(21,44)(22,54)(23,45)(24,46)(25,55)(26,57)(27,50)(28,48)(29,49)(34,58)(35,61)(36,43)(37,62)(38,42)(39,60)(47,52)(51,59)(53,64)(56,63);(1,4,7,23,8,24,6,22)(2,10,13,20,14,36,12,35)(3,16,18,45,19,46,30,54)(5,26,28,21,29,52,27,53)(9,31,15,41,33,43,40,61)(11,37,39,34,25,51,38,56)(17,47,48,64,49,57,50,44)(32,59,60,63,55,62,42,58);(1,5)(2,11)(3,17)(4,21)(6,28)(7,27)(8,29)(9,32)(10,34)(12,39)(13,38)(14,25)(15,42)(16,44)(18,50)(19,49)(20,37)(22,52)(23,26)(24,53)(30,48)(31,58)(33,55)(35,51)(36,56)(40,60)(41,62)(43,63)(45,57)(46,64)(47,54)(59,61)

Polycyclic presentation:

2^9|a^2=g,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, 368620);