#507245

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	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

546 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)(19,23)(20,24)(21,25)(22,26)(27,42)(28,35)(29,36)(30,37)(31,38)(32,39)(33,41)(34,43)(40,46)(44,47)(45,48);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,21,29,32,45,46,43,20,28,31,44,22,30,33,27)(18,23,25,36,39,48,40,34,24,35,38,47,26,37,41,42);(10,13)(14,15)(18,24)(19,27)(21,33)(22,32)(23,34)(25,40)(28,43)(29,30)(31,46)(35,42)(36,48)(37,47)(38,41)(44,45);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(19,28)(23,35)(27,43)(29,44)(30,45)(34,42)(36,47)(37,48)

Polycyclic presentation:

2^9|b^2=efgi,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,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, 414441);