#8728415

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

4115 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8635611);