#10188285

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	#36814	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,11)(4,12)(5,9)(6,14)(7,8)(10,13)(15,16)(17,18,22,25)(19,23,29,32)(20,24,30,21)(26,28,27,31)(33,34,39,37)(35,43,45,40)(36,44,46,41)(38,42,48,47);(1,3,7,12)(2,4,8,11)(5,13,15,6)(9,14,16,10)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,32)(26,30)(33,35)(34,40)(36,42)(37,43)(38,44)(39,45)(41,48)(46,47);(2,8)(4,11)(9,16)(10,14)(17,20)(18,24)(19,26)(21,25)(22,30)(23,28)(27,29)(31,32)(33,34,39,37)(35,40,45,43)(36,41,46,44)(38,42,48,47);(1,4,7,11)(2,3,8,12)(5,14,15,10)(6,9,13,16)(17,20)(18,24)(19,27)(21,25)(22,30)(23,31)(26,29)(28,32)(33,36)(34,41)(35,42)(37,44)(38,43)(39,46)(40,48)(45,47);(1,5,7,15)(2,9,8,16)(3,6,12,13)(4,10,11,14)(17,19)(18,23)(20,26)(21,31)(22,29)(24,28)(25,32)(27,30)(33,37,39,34)(35,43,45,40)(36,41,46,44)(38,42,48,47);(1,6)(2,10)(3,5)(4,9)(7,13)(8,14)(11,16)(12,15)(17,21,22,24)(18,20,25,30)(19,28,29,31)(23,27,32,26)(33,38,39,48)(34,42,37,47)(35,44,45,41)(36,43,46,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10095481);