#9058572

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	#46661	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965768);