#10247195

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(4,12)(7,9)(10,13)(17,18,23,28)(19,26,29,24)(20,25,21,30)(22,27,32,31);(1,2,5,6)(3,9,11,7)(4,8,12,14)(10,16,13,15)(17,19)(18,24)(20,27)(21,31)(22,25)(23,29)(26,28)(30,32)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(1,2,5,6)(3,7,11,9)(4,8,12,14)(10,15,13,16)(17,20,23,21)(18,25,28,30)(19,27,29,31)(22,26,32,24)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,21,23,20)(18,25,28,30)(19,27,29,31)(22,24,32,26)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,3,5,11)(2,7,6,9)(4,13,12,10)(8,16,14,15)(17,19)(18,26)(20,27)(21,31)(22,30)(23,29)(24,28)(25,32)(33,35)(34,36)(37,45)(38,43)(39,41)(40,47)(42,46)(44,48);(1,4,5,12)(2,8,6,14)(3,10,11,13)(7,15,9,16)(17,22,23,32)(18,27,28,31)(19,25,29,30)(20,26,21,24)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154391);