#8734383

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	#52472	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,f^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, 8641579);