#8097091

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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)(3,9,4,8)(5,6,10,16)(12,15,14,13)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,3,7,4)(2,8,11,9)(5,12,10,14)(6,13,16,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(36,44)(39,46)(43,47)(45,48);(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,45,46,48);(1,4,7,3)(2,9,11,8)(5,14,10,12)(6,15,16,13)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,46)(42,48);(1,5)(2,6)(3,12)(4,14)(7,10)(8,13)(9,15)(11,16)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48);(1,6)(2,10)(3,13)(4,15)(5,11)(7,16)(8,14)(9,12)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=i,f^2=i,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, 8004287);