#10247190

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	#40648	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3862 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=hi,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, 10154386);