#8751969

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	#55737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 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)(3,9)(4,6)(5,8)(7,10)(11,15)(12,13)(14,16)(19,25)(23,26)(27,31)(30,32)(33,34)(35,37)(36,43)(38,39)(40,45)(41,46)(42,48)(44,47);(2,4)(5,13)(7,9)(11,16)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,34)(35,37)(36,40)(38,39)(41,46)(42,44)(43,45)(47,48);(3,10)(7,9)(11,16)(14,15)(18,22)(23,26)(24,28)(30,32)(34,39)(35,41)(40,43)(42,47);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,43)(41,47)(46,48);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8659165);