#10247501

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

(1,2,7,4)(3,5,8,11)(6,12,15,9)(10,14,16,13)(17,18,24,27)(19,23,22,25)(20,26,21,31)(28,30,29,32);(1,3)(2,5)(4,11)(6,10)(7,8)(9,13)(12,14)(15,16)(17,19,24,22)(18,25,27,23)(20,28,21,29)(26,32,31,30)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(2,4)(3,8)(9,12)(10,16)(17,20,24,21)(18,26,27,31)(19,28,22,29)(23,30,25,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,4,7,2)(3,5,8,11)(6,12,15,9)(10,13,16,14)(17,21,24,20)(18,26,27,31)(19,28,22,29)(23,32,25,30)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,5)(2,8)(3,4)(6,14)(7,11)(9,10)(12,16)(13,15)(17,22,24,19)(18,25,27,23)(20,29,21,28)(26,32,31,30)(33,35)(34,36)(37,45)(38,43)(39,41)(40,47)(42,46)(44,48);(1,6,7,15)(2,9,4,12)(3,10,8,16)(5,13,11,14)(17,23,24,25)(18,22,27,19)(20,30,21,32)(26,29,31,28)(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=g,d^2=gi,e^2=g,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, 10154697);