#8210202

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 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:

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

Polycyclic presentation:

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

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, 8117398);