#477835

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17124
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

679 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385031);