#507251

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

515 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)(19,24)(20,33)(21,26)(22,27)(23,28)(25,35)(29,39)(30,44)(31,40)(32,41)(34,45)(36,46)(37,48)(38,42)(43,47);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,29,37,30,35,38,36,21,31,22,32,23,20,34,47)(18,24,39,48,44,25,42,46,26,40,27,41,28,33,45,43);(10,13)(14,15)(17,20,21,35)(18,25,26,33)(19,30,31,23)(22,37,29,32)(24,28,40,44)(27,41,39,48)(34,47,38,36)(42,43,45,46);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(18,26)(24,40)(25,33)(27,39)(28,44)(41,48)(42,45)(43,46)

Polycyclic presentation:

2^9|b^2=efg,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 414447);