#10040592

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	#2703	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)(17,18,23,25)(19,24,29,32)(20,30,31,26)(21,27,28,22)(33,34)(35,37)(36,41)(38,40)(39,42)(43,44)(45,48)(46,47);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,19)(18,24)(20,22)(21,30)(23,29)(25,32)(26,28)(27,31)(33,35,39,38)(34,40,42,37)(36,43,45,46)(41,47,48,44);(2,7)(4,10)(9,13)(12,14)(18,25)(19,29)(20,31)(21,28)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,16)(13,15)(17,20)(18,26)(19,27)(21,24)(22,29)(23,31)(25,30)(28,32)(33,37,39,40)(34,35,42,38)(36,44,45,47)(41,43,48,46);(1,4,6,10)(2,8,7,3)(5,14,15,12)(9,11,13,16)(17,21)(18,27)(19,26)(20,24)(22,25)(23,28)(29,30)(31,32)(33,38,39,35)(34,40,42,37)(36,43,45,46)(41,44,48,47);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,22,23,27)(18,28,25,21)(19,20,29,31)(24,26,32,30)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=ghi,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, 9947788);