#8934816

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,7)(4,6)(5,8)(9,11)(10,14)(12,13)(15,16)(17,18)(19,24)(20,26)(21,29)(22,23)(25,30)(27,28)(31,32)(33,34,39,42)(35,40,43,48)(36,46,45,38)(37,47,44,41);(2,4)(3,9)(8,12)(10,15)(18,23)(19,26)(25,29)(27,32)(33,35)(34,40)(36,37)(38,41)(39,43)(42,48)(44,45)(46,47);(3,9)(5,13)(7,11)(8,12)(18,23)(19,26)(21,30)(28,31)(35,43)(36,45)(38,46)(40,48);(1,3,6,9)(2,7,4,11)(5,10,13,15)(8,14,12,16)(17,19,22,26)(18,24,23,20)(21,27,30,32)(25,31,29,28)(33,36)(34,38)(35,44)(37,43)(39,45)(40,47)(41,48)(42,46);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,20,22,24)(18,19,23,26)(21,28,30,31)(25,27,29,32)(33,37,39,44)(34,41,42,47)(35,45,43,36)(38,40,46,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,30)(18,25,23,29)(19,27,26,32)(20,28,24,31)(33,38)(34,36)(35,41)(37,40)(39,46)(42,45)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=gi,f^2=h,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, 8842012);