#472665

order	512
nilpotency class	3
isoclinism	isoclinic class of #457608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2735
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16830
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

710 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fgh,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,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, 379861);