#8720716

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
prime radical	#46575	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,28,31,32)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,48,45);(2,7)(3,10)(9,12)(11,16)(17,19,21,26)(18,22,24,29)(20,25,27,31)(23,28,30,32)(33,35,39,37)(34,40,44,42)(36,43,47,45)(38,41,48,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,39,37)(34,40,44,42)(36,43,47,45)(38,41,48,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,31)(30,32)(33,36,39,47)(34,41,44,46)(35,45,37,43)(38,42,48,40);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37,39,35)(34,42,44,40)(36,43,47,45)(38,41,48,46);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32)(33,38,39,48)(34,43,44,45)(35,41,37,46)(36,40,47,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627912);