#6611342

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,18,23,27)(19,28,32,42)(20,33,34,25)(21,26,22,37)(24,30,39,31)(29,43,44,41)(35,48,36,40)(38,46,47,45);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,30,23,32,22,31)(18,24,37,42,27,39,26,28)(20,29,35,45,34,44,36,46)(25,38,48,43,33,47,40,41);(1,3,4,8)(2,6,5,7)(11,16)(13,14)(17,20,23,34)(18,25,27,33)(19,29,32,44)(21,35,22,36)(24,38,39,47)(26,40,37,48)(28,41,42,43)(30,45,31,46);(9,11)(10,14)(12,16)(13,15)(20,34)(25,33)(29,44)(35,36)(38,47)(40,48)(41,43)(45,46);(11,16)(13,14)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,34,36)(24,28,39,42)(25,40,33,48)(29,45,44,46)(38,41,47,43)

Polycyclic presentation:

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

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, 6518538);