#8633957

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	#55697	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,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, 8541153);