#8534523

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3818 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441719);