#10247196

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	#42018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,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, 10154392);