#6475916

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

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, 6383112);