#8934853

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

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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,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, 8842049);