#10195621

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	#51166	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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)(3,13)(4,10)(5,6)(7,11)(8,12)(9,15)(14,16)(17,18,22,25)(19,24,28,23)(20,21,31,29)(26,32,30,27)(33,34,39,43)(35,40,45,38)(36,41,44,48)(37,42,46,47);(1,3)(2,9)(4,7)(5,14)(6,11)(8,15)(10,16)(12,13)(17,18,22,25)(19,24,28,23)(20,29,31,21)(26,27,30,32)(33,35)(34,40)(36,46)(37,44)(38,43)(39,45)(41,47)(42,48);(1,4,8,5)(2,10,12,6)(3,7,15,14)(9,16,13,11)(17,19,22,28)(18,23,25,24)(20,26,31,30)(21,27,29,32)(33,36,39,44)(34,41,43,48)(35,37,45,46)(38,47,40,42);(1,5,8,4)(2,10,12,6)(3,14,15,7)(9,16,13,11)(17,19)(18,24)(20,30)(21,27)(22,28)(23,25)(26,31)(29,32)(33,35)(34,40)(36,37)(38,43)(39,45)(41,42)(44,46)(47,48);(1,6,8,10)(2,5,12,4)(3,11,15,16)(7,9,14,13)(17,20,22,31)(18,21,25,29)(19,26,28,30)(23,27,24,32)(33,37)(34,42)(35,44)(36,45)(38,41)(39,46)(40,48)(43,47);(1,7,8,14)(2,11,12,16)(3,4,15,5)(6,13,10,9)(17,21)(18,20)(19,27)(22,29)(23,26)(24,30)(25,31)(28,32)(33,38,39,40)(34,35,43,45)(36,47,44,42)(37,48,46,41)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=g,e^2=gh,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, 10102817);