#10038438

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gi,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, 9945634);