#8475825

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4217 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8383021);