#10247336

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	#54957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154532);