#10040706

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3894 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=hi,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, 9947902);