#6495595

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6402791);