#441611

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

975 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)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,33)(21,28)(22,25)(23,27)(24,31)(26,37)(30,40)(32,44)(34,45)(35,41)(36,43)(38,39)(42,47)(46,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,32)(18,24,25,39)(20,34,36,47)(21,38,23,31)(26,42,41,45)(27,44,28,29)(30,37,46,35)(33,48,43,40);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(18,25)(24,39)(26,41)(27,28)(29,44)(33,43)(40,48)(42,45);(17,20,22,36)(18,26,25,41)(19,30,32,46)(21,37,23,35)(24,40,39,48)(27,33,28,43)(29,45,44,42)(31,47,38,34)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348807);