#8734709

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

4239 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18,21,22)(19,25,27,23)(20,24,30,28)(26,32,29,31)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(2,7)(3,11)(9,12)(10,16)(17,18,21,22)(19,23,27,25)(20,24,30,28)(26,31,29,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,48)(45,47);(2,7)(3,11)(9,12)(10,16)(18,22)(23,25)(24,28)(31,32)(36,44)(40,45)(42,48)(46,47);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,36,38,44)(34,40,37,45)(35,42,43,48)(39,46,41,47);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,48)(45,46);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,37,38,34)(35,41,43,39)(36,40,44,45)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=g,e^2=h,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, 8641905);