#10042597

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3896 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,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, 9949793);