#8934242

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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)(4,12)(5,10)(6,7)(9,13)(11,15)(14,16)(17,18,23,24)(19,25,21,20)(22,30,31,26)(27,28,29,32)(33,34,38,42)(35,39,43,47)(36,45,46,40)(37,44,48,41);(2,7)(3,10)(4,13)(15,16)(17,19,23,21)(18,20,24,25)(22,27,31,29)(26,28,30,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,48)(42,47)(44,46);(3,10)(5,8)(11,14)(15,16)(18,24)(20,25)(22,31)(27,29)(35,43)(36,46)(39,47)(40,45);(1,3,6,10)(2,8,7,5)(4,14,13,11)(9,15,12,16)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,30)(31,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,4)(2,9)(3,11)(5,15)(6,13)(7,12)(8,16)(10,14)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,37)(34,41)(35,40)(36,47)(38,48)(39,46)(42,44)(43,45);(1,5,6,8)(2,10,7,3)(4,15,13,16)(9,14,12,11)(17,22,23,31)(18,26,24,30)(19,27,21,29)(20,28,25,32)(33,35)(34,39)(36,44)(37,40)(38,43)(41,46)(42,47)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gh,e^2=h,f^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, 8841438);