#8374205

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

3926 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8281401);