#10040774

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3799 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)(3,8,11,15)(4,14,13,9)(6,16,12,10)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(33,34,38,39)(35,40,45,48)(36,41,46,44)(37,42,47,43);(1,3,7,11)(2,8,5,15)(4,6,13,12)(9,10,14,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(34,39)(36,46)(40,48)(42,43);(2,5)(3,11)(4,13)(10,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,35,38,45)(34,40,39,48)(36,43,46,42)(37,44,47,41);(1,4,7,13)(2,9,5,14)(3,12,11,6)(8,16,15,10)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36)(34,41)(35,43)(37,40)(38,46)(39,44)(42,45)(47,48);(1,5,7,2)(3,8,11,15)(4,14,13,9)(6,10,12,16)(17,20,22,18)(19,23,27,24)(21,25,29,28)(26,31,32,30)(33,35)(34,40)(36,43)(37,41)(38,45)(39,48)(42,46)(44,47);(1,6,7,12)(2,10,5,16)(3,13,11,4)(8,14,15,9)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,37)(34,42)(35,44)(36,40)(38,47)(39,43)(41,45)(46,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947970);