#10040782

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3863 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)(18,22)(20,28)(23,25)(26,32)(33,34,39,37)(35,42,38,40)(36,41,44,46)(43,48,45,47);(1,3,7,11)(2,8,5,15)(4,6,13,12)(9,10,14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(34,37)(36,44)(40,42)(43,45);(2,5)(3,11)(4,13)(10,16)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,35,39,38)(34,40,37,42)(36,45,44,43)(41,48,46,47);(1,4,7,13)(2,9,5,14)(3,12,11,6)(8,16,15,10)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,39,44)(34,41,37,46)(35,43,38,45)(40,47,42,48);(1,5,7,2)(3,8,11,15)(4,14,13,9)(6,10,12,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37,39,34)(35,40,38,42)(36,46,44,41)(43,47,45,48);(1,6,7,12)(2,10,5,16)(3,13,11,4)(8,14,15,9)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38,39,35)(34,42,37,40)(36,45,44,43)(41,48,46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=gi,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, 9947978);