#8933126

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	#48995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(3,9)(4,7)(5,13)(6,8)(10,14)(11,15)(12,16)(19,25)(20,28)(23,26)(24,29)(33,34,38,39)(35,37,41,48)(36,43,45,40)(42,46,44,47);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,35)(34,37)(36,44)(38,41)(39,48)(40,46)(42,45)(43,47);(2,8)(5,14)(7,9)(11,12)(18,22)(19,25)(24,29)(27,30)(34,39)(35,41)(36,45)(46,47);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,36,38,45)(34,40,39,43)(35,42,41,44)(37,47,48,46);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,37)(34,35)(36,46)(38,48)(39,41)(40,44)(42,43)(45,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,34)(35,37)(36,40)(38,39)(41,48)(42,47)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8840322);