#6495995

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1493 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18,22,26)(19,27,32,44)(20,24,33,40)(21,35,28,25)(23,36,39,31)(29,42,30,43)(34,48,45,41)(37,47,38,46);(9,10)(11,14)(12,13)(15,16)(17,19,28,36,22,32,21,31)(18,23,35,44,26,39,25,27)(20,29,45,47,33,30,34,46)(24,37,48,43,40,38,41,42);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,20)(18,24)(19,29)(21,34)(22,33)(23,37)(25,41)(26,40)(27,42)(28,45)(30,32)(31,46)(35,48)(36,47)(38,39)(43,44);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,24)(19,30)(21,34)(22,33)(23,38)(25,41)(26,40)(27,43)(28,45)(29,32)(31,47)(35,48)(36,46)(37,39)(42,44);(9,11,12,15)(10,14,13,16)(20,33)(24,40)(29,30)(34,45)(37,38)(41,48)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,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, 6403191);