#10040458

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3803 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3,6,11)(2,7,8,16)(4,14,10,12)(5,13,9,15)(17,18)(19,23)(20,22)(21,25)(24,27)(26,30)(28,29)(31,32)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(2,8)(3,11)(4,10)(13,15)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,35,38,43)(34,39,42,46)(36,44,45,37)(40,47,48,41);(1,4,6,10)(2,9,8,5)(3,12,11,14)(7,13,16,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3)(2,7)(4,12)(5,15)(6,11)(8,16)(9,13)(10,14)(17,20,22,18)(19,23,27,24)(21,25,29,28)(26,31,32,30)(33,36,38,45)(34,40,42,48)(35,37,43,44)(39,41,46,47);(1,5,6,9)(2,10,8,4)(3,13,11,15)(7,12,16,14)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,37)(34,41)(35,36)(38,44)(39,40)(42,47)(43,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947654);