#8810193

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	#56739	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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)(3,5)(4,9)(6,7)(8,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,24,27,21)(20,25,30,31)(26,32,28,29)(34,36)(35,40)(38,46)(45,48);(2,7)(4,12)(5,8)(11,16)(18,23)(21,24)(25,31)(29,32)(33,34,39,36)(35,42,40,37)(38,41,46,44)(43,48,47,45);(2,7)(3,10)(4,12)(13,15)(17,19,22,27)(18,24,23,21)(20,28,30,26)(25,29,31,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,15,14,13)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,36,39,34)(35,42,40,37)(38,44,46,41)(43,48,47,45);(1,4,6,12)(2,9,7,14)(3,11,10,16)(5,13,8,15)(17,20)(18,25)(19,26)(21,29)(22,30)(23,31)(24,32)(27,28)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,5,6,8)(2,10,7,3)(4,13,12,15)(9,16,14,11)(17,21)(18,19)(20,29)(22,24)(23,27)(25,26)(28,31)(30,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717389);