#8712762

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

3979 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=gh,e^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, 8619958);