#8813952

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8721148);