#8188521

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095717);