#10195850

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,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, 10103046);