#147992

order	512
nilpotency class	3
isoclinism	isoclinic class of #2069
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#396
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11042
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55188);