#508650

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,24,28,38)(20,30,32,25)(22,26,34,37)(27,43,44,39)(29,35,41,47)(31,45,48,40)(33,46,42,36);(2,5)(6,8)(17,19,21,28)(18,24,23,38)(20,31,32,48)(22,35,34,47)(25,40,30,45)(26,41,37,29)(27,42,44,33)(36,39,46,43);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,20,21,32)(18,25,23,30)(19,27,28,44)(22,36,34,46)(24,39,38,43)(26,42,37,33)(29,45,41,40)(31,47,48,35);(3,7)(6,8)(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415846);