#8197105

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,4)(5,8)(6,14)(7,10)(9,15)(11,13)(12,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34,39,38)(35,40,45,44)(36,46,41,37)(42,48,47,43);(1,3,7,8)(2,4,10,5)(6,11,15,16)(9,12,14,13)(18,23)(19,26)(25,28)(27,32)(33,35,39,45)(34,40,38,44)(36,42,41,47)(37,43,46,48);(1,4,7,5)(2,8,10,3)(6,12,15,13)(9,16,14,11)(17,18,22,23)(19,24,26,21)(20,25,30,28)(27,31,32,29)(33,36)(34,37)(35,42)(38,46)(39,41)(40,43)(44,48)(45,47);(1,5,7,4)(2,8,10,3)(6,13,15,12)(9,16,14,11)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,35)(34,40)(36,42)(37,43)(38,44)(39,45)(41,47)(46,48);(1,6,7,15)(2,9,10,14)(3,11,8,16)(4,12,5,13)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,37,39,46)(34,41,38,36)(35,43,45,48)(40,47,44,42);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,38,39,34)(35,44,45,40)(36,46,41,37)(42,48,47,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8104301);