#505101

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,28)(19,24,32,40)(20,25,22,27)(21,26,35,43)(29,37,46,48)(30,38,36,44)(31,39,45,47)(33,41,34,42);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19)(18,24)(20,30)(21,29)(22,36)(23,32)(25,38)(26,37)(27,44)(28,40)(31,34)(33,45)(35,46)(39,42)(41,47)(43,48);(10,13)(14,15)(19,29)(20,22)(21,34)(24,37)(25,27)(26,42)(30,31)(32,46)(33,35)(36,45)(38,39)(40,48)(41,43)(44,47);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(17,20,23,22)(18,25,28,27)(19,30,32,36)(21,33,35,34)(24,38,40,44)(26,41,43,42)(29,45,46,31)(37,47,48,39)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi

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, 412297);