#508507

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=f,c^2=i,d^2=i,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, 415703);