#6484420

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469983
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×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6391616);