#8188569

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=ghi,e^2=h,f^2=g,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, 8095765);