#8096853

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004049);