#10247625

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

3786 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:

(1,2,6,5)(3,9,11,7)(4,8,13,14)(10,16,12,15)(18,23)(19,27)(21,30)(28,31);(1,3,6,11)(2,7,5,9)(4,10,13,12)(8,15,14,16)(17,18)(19,24)(20,30)(21,26)(22,23)(25,27)(28,32)(29,31)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(17,19,22,27)(18,24,23,25)(20,28,26,31)(21,29,30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,13)(14,15)(17,19)(18,25)(20,31)(21,29)(22,27)(23,24)(26,28)(30,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,4)(2,8)(3,10)(5,14)(6,13)(7,15)(9,16)(11,12)(17,20)(18,21)(19,28)(22,26)(23,30)(24,29)(25,32)(27,31)(33,35)(34,36)(37,45)(38,43)(39,41)(40,47)(42,46)(44,48);(1,5,6,2)(3,9,11,7)(4,8,13,14)(10,15,12,16)(17,21)(18,26)(19,29)(20,23)(22,30)(24,31)(25,28)(27,32)(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=ghi,c^2=h,d^2=i,f^2=gh,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, 10154821);