#8107601

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014797);