#8581506

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=hi,e^2=ghi,f^2=i,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, 8488702);