#6435220

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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

1556 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi,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, 6342416);