#8605709

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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)(3,4)(5,13)(6,10)(7,8)(9,16)(11,14)(12,15)(17,18,24,29)(19,22,31,25)(20,27,21,26)(23,28,30,32);(1,3,7,11)(2,4,8,14)(5,15,16,6)(9,10,13,12)(17,19)(18,25)(20,30)(21,23)(22,29)(24,31)(26,32)(27,28)(33,34,38,41)(35,42,45,39)(36,37,40,46)(43,48,47,44);(2,8)(3,11)(5,16)(10,12)(17,20,24,21)(18,26,29,27)(19,30,31,23)(22,28,25,32)(33,35,38,45)(34,39,41,42)(36,43,40,47)(37,44,46,48);(1,4)(2,3)(5,12)(6,9)(7,14)(8,11)(10,16)(13,15)(17,21,24,20)(18,27,29,26)(19,30,31,23)(22,28,25,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,5)(2,9)(3,6)(4,12)(7,16)(8,13)(10,14)(11,15)(17,22)(18,19)(20,32)(21,28)(23,26)(24,25)(27,30)(29,31)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(1,6)(2,10)(3,5)(4,13)(7,15)(8,12)(9,14)(11,16)(17,23,24,30)(18,28,29,32)(19,21,31,20)(22,26,25,27)(33,37)(34,40)(35,44)(36,41)(38,46)(39,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=g,f^2=g,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, 8512905);