#8741691

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	#38509	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4116 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8648887);