#9059206

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	#43204	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30);(1,3,7,5)(2,8,9,11)(4,12,14,13)(6,10,16,15)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,47)(46,48);(2,9)(3,5)(4,14)(6,16)(19,24)(20,23)(21,29)(25,32)(34,38)(35,42)(36,45)(47,48);(1,4,7,14)(2,10,9,15)(3,13,5,12)(6,11,16,8)(17,19)(18,23)(20,26)(21,27)(22,24)(25,30)(28,32)(29,31)(33,35)(34,39)(36,46)(37,42)(38,41)(40,48)(43,45)(44,47);(1,5,7,3)(2,11,9,8)(4,13,14,12)(6,15,16,10)(17,20,22,23)(18,24,26,19)(21,28,29,30)(25,31,32,27)(33,36)(34,40)(35,43)(37,45)(38,44)(39,47)(41,48)(42,46);(1,6)(2,12)(3,10)(4,11)(5,15)(7,16)(8,14)(9,13)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=gh,f^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, 8966402);