#8942224

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	#41976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8849420);