#10040514

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3809 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,34,39,43)(35,41,45,37)(36,40,46,48)(38,42,47,44);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(19,26)(20,29)(23,25)(24,28)(33,35,39,45)(34,37,43,41)(36,44,46,42)(38,40,47,48);(2,7)(4,10)(9,13)(12,14)(17,19,22,26)(18,23,21,25)(20,30,29,27)(24,32,28,31)(33,36,39,46)(34,40,43,48)(35,44,45,42)(37,47,41,38);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,16)(13,15)(17,20,22,29)(18,24,21,28)(19,27,26,30)(23,31,25,32)(33,35)(34,41)(36,44)(37,43)(38,40)(39,45)(42,46)(47,48);(1,4,6,10)(2,8,7,3)(5,14,15,12)(9,11,13,16)(17,19)(18,25)(20,27)(21,23)(22,26)(24,32)(28,31)(29,30)(33,37,39,41)(34,35,43,45)(36,38,46,47)(40,42,48,44);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,21,22,18)(19,25,26,23)(20,28,29,24)(27,32,30,31)(33,38)(34,42)(35,40)(36,37)(39,47)(41,46)(43,44)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=i,e^2=g,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, 9947710);