#8603505

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8510701);