#8772694

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

4240 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)(5,13)(6,7)(8,11)(9,14)(10,15)(12,16)(17,18,22,21)(19,23,27,26)(20,28,30,24)(25,32,29,31)(34,37)(39,41)(40,45)(46,47);(2,7)(4,8)(5,14)(10,16)(18,21)(23,26)(24,28)(31,32)(34,37)(36,44)(39,41)(42,48);(1,3,6,11)(2,8,7,4)(5,10,14,16)(9,15,13,12)(17,19,22,27)(18,23,21,26)(20,29,30,25)(24,32,28,31)(33,34,38,37)(35,41,43,39)(36,45,44,40)(42,46,48,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,13)(14,16)(17,20,22,30)(18,24,21,28)(19,25,27,29)(23,31,26,32)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,36,38,44)(34,40,37,45)(35,42,43,48)(39,46,41,47);(1,5)(2,9)(3,10)(4,12)(6,14)(7,13)(8,15)(11,16)(17,21,22,18)(19,26,27,23)(20,28,30,24)(25,32,29,31)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8679890);