#471499

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	#16798
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

893 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 378695);