#507257

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 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)(19,27)(20,24)(21,25)(22,26)(23,29)(28,35)(30,43)(31,44)(32,38)(33,39)(34,41)(36,46)(37,47)(40,48)(42,45);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,21,30,33,47,48,45,20,29,32,46,22,31,34,28)(18,23,25,36,39,44,40,35,24,27,38,43,26,37,41,42);(10,13)(14,15)(18,24)(19,28)(21,34)(22,33)(23,35)(25,40)(27,42)(29,45)(30,31)(32,48)(36,44)(37,43)(38,41)(46,47);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(19,29)(23,27)(28,45)(30,46)(31,47)(35,42)(36,43)(37,44)

Polycyclic presentation:

2^9|b^2=efg,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,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, 414453);