#8934307

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	#35421	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=h,e^2=gh,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, 8841503);