#8633891

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8541087);