#8617761

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 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)(3,9)(4,11)(5,8)(6,7)(10,16)(12,13)(14,15)(17,18,23,27)(19,24,21,25)(20,22,26,32)(28,29,30,31)(33,34,39,43)(35,40,46,45)(36,47,37,41)(38,42,48,44);(1,3,6,11)(2,4,7,9)(5,10,14,13)(8,12,15,16)(17,19)(18,24)(20,28)(21,23)(22,29)(25,27)(26,30)(31,32)(33,35,39,46)(34,40,43,45)(36,44,37,42)(38,41,48,47);(2,7)(4,9)(8,15)(12,16)(19,21)(24,25)(28,30)(29,31)(33,36,39,37)(34,41,43,47)(35,42,46,44)(38,45,48,40);(1,3)(2,4)(5,13)(6,11)(7,9)(8,16)(10,14)(12,15)(17,20)(18,22)(19,28)(21,30)(23,26)(24,29)(25,31)(27,32)(33,37,39,36)(34,41,43,47)(35,44,46,42)(38,45,48,40);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21,23,19)(18,25,27,24)(20,30,26,28)(22,31,32,29)(33,34)(35,45)(36,47)(37,41)(38,42)(39,43)(40,46)(44,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,22)(18,26)(19,29)(20,27)(21,31)(23,32)(24,30)(25,28)(33,38,39,48)(34,42,43,44)(35,41,46,47)(36,45,37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8524957);