#8180361

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	#40414	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087557);