#8661361

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

4193 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)(3,13)(4,10)(5,7)(6,11)(8,12)(9,16)(14,15)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34,39,40)(35,44,45,41)(36,48,42,37)(38,43,46,47);(1,3,8,16)(2,9,12,13)(4,14,5,6)(7,15,10,11)(18,20)(23,25)(24,28)(31,32)(34,40)(35,45)(36,42)(43,47);(1,4,8,5)(2,10,12,7)(3,14,16,6)(9,11,13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,5,8,4)(2,10,12,7)(3,6,16,14)(9,11,13,15)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,36,39,42)(34,37,40,48)(35,46,45,38)(41,47,44,43);(1,6)(2,11)(3,4)(5,16)(7,9)(8,14)(10,13)(12,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,37)(34,42)(35,47)(36,40)(38,41)(39,48)(43,45)(44,46);(1,7,8,10)(2,5,12,4)(3,15,16,11)(6,13,14,9)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,38,39,46)(34,43,40,47)(35,42,45,36)(37,41,48,44)

Polycyclic presentation:

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

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, 8568557);