#8180498

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18,22,23)(19,25,27,21)(20,28,30,24)(26,29,32,31)(35,41)(36,44)(39,42)(40,45);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,31)(30,32)(33,34,38,37)(35,42,41,39)(36,40,44,45)(43,48,47,46);(6,14)(9,12)(11,13)(15,16)(18,23)(21,25)(24,28)(29,31)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,47)(45,48);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20,22,30)(18,24,23,28)(19,26,27,32)(21,29,25,31);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,31)(30,32)(33,36,38,44)(34,40,37,45)(35,43,41,47)(39,46,42,48);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,21)(18,19)(20,29)(22,25)(23,27)(24,26)(28,32)(30,31)(33,37,38,34)(35,39,41,42)(36,45,44,40)(43,46,47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^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, 8087694);