#494022

order	512
nilpotency class	4
isoclinism	isoclinic class of #480315
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#20723
prime radical	#3455	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #97 ◁ #2875 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

732 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)(3,13,14,40)(4,10)(5,11)(6,24,25,9)(7,12)(8,16,29,41)(15,38,18,46)(17,37)(19,35)(20,34)(21,33,50,30)(22,52,53,31)(23,36)(26,54)(27,43,32,47)(28,45)(39,42)(44,60)(48,61)(49,59)(51,58,64,57)(55,63)(56,62);(1,3)(2,8,12,29)(4,18)(5,16)(6,17)(7,14)(9,28,24,26)(10,32,35,27)(11,13,36,40)(15,19)(20,47)(21,48)(22,45)(23,41)(25,42)(30,55,33,56)(31,39,52,37)(34,46,59,38)(43,49)(44,50)(51,63)(53,54)(57,60,58,61)(62,64);(2,9)(3,14)(4,19)(8,26)(10,33)(11,31)(12,24)(13,37)(16,41)(17,42)(20,49)(21,50)(27,56)(28,29)(30,35)(32,55)(34,58)(36,52)(38,61)(39,40)(45,54)(46,60)(51,64)(57,59);(1,4,5,20,7,19,23,49)(2,10,11,34,12,35,36,59)(3,15,16,43,14,18,41,47)(6,21,22,51,25,50,53,64)(8,27,13,38,29,32,40,46)(9,30,31,57,24,33,52,58)(17,44,45,62,42,48,54,63)(26,55,37,60,28,56,39,61)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=e,e^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=di,g^b=gi,d^c=di,g^c=gh,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401218);