#10159194

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3833 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,e^2=hi,f^2=g,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, 10066390);