#8734725

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

4147 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18,21,22)(19,25,27,23)(20,24,30,28)(26,32,29,31)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,48)(46,47);(2,7)(3,8)(9,16)(10,13)(17,18,21,22)(19,23,27,25)(20,24,30,28)(26,31,29,32)(33,35,38,43)(34,39,42,48)(36,37,44,47)(40,46,45,41);(2,7)(3,8)(9,16)(10,13)(18,22)(23,25)(24,28)(31,32)(35,43)(36,44)(39,48)(40,45);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,36)(34,40)(35,37)(38,44)(39,46)(41,48)(42,45)(43,47);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,48)(44,45);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,37)(34,41)(35,44)(36,43)(38,47)(39,40)(42,46)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=g,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, 8641921);