#10195855

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

3846 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)(3,9)(4,14)(5,7)(6,10)(8,11)(12,15)(13,16)(17,18,23,21)(19,27,29,24)(20,28,30,25)(22,26,32,31)(33,34,39,43)(35,40,45,38)(36,41,44,48)(37,42,46,47);(1,3,8,5)(2,9,11,7)(4,13,6,12)(10,15,14,16)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,32)(30,31)(33,35)(34,40)(36,46)(37,44)(38,43)(39,45)(41,47)(42,48);(1,4,8,6)(2,10,11,14)(3,12,5,13)(7,15,9,16)(17,18,23,21)(19,24,29,27)(20,25,30,28)(22,26,32,31)(33,36,39,44)(34,41,43,48)(35,37,45,46)(38,47,40,42);(1,5,8,3)(2,7,11,9)(4,12,6,13)(10,16,14,15)(17,20)(18,25)(19,26)(21,28)(22,27)(23,30)(24,32)(29,31)(33,35)(34,40)(36,37)(38,43)(39,45)(41,42)(44,46)(47,48);(1,6,8,4)(2,10,11,14)(3,13,5,12)(7,15,9,16)(17,21,23,18)(19,27,29,24)(20,25,30,28)(22,26,32,31)(33,37)(34,42)(35,44)(36,45)(38,41)(39,46)(40,48)(43,47);(1,7)(2,3)(4,15)(5,11)(6,16)(8,9)(10,12)(13,14)(17,22,23,32)(18,26,21,31)(19,28,29,25)(20,27,30,24)(33,38,39,40)(34,35,43,45)(36,47,44,42)(37,48,46,41)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=h,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, 10103051);