#8827860

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	#46320	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8735056);