#8617795

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

3946 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=hi,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, 8524991);