#8621811

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=h,e^2=hi,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, 8529007);