#6495048

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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

1384 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)(17,18,22,26)(19,28,31,45)(20,25,33,24)(21,35,36,27)(23,37,40,30)(29,44,32,43)(34,41,46,42)(38,47,39,48);(9,10)(11,14)(12,13)(15,16)(17,19,21,30,22,31,36,37)(18,23,27,28,26,40,35,45)(20,32,34,48,33,29,46,47)(24,39,41,44,25,38,42,43);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(17,20)(18,24)(19,29)(21,34)(22,33)(23,38)(25,26)(27,41)(28,43)(30,47)(31,32)(35,42)(36,46)(37,48)(39,40)(44,45);(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,39)(24,26)(27,42)(28,44)(30,47)(31,32)(35,41)(36,46)(37,48)(38,40)(43,45);(9,11,12,15)(10,14,13,16)(18,26)(20,33)(23,40)(27,35)(28,45)(29,32)(34,46)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,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^c=ei,e^d=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, 6402244);