#8096946

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	#45737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004142);