#8718094

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

3952 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625290);