#8097059

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	#52695	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,e^2=gh,f^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, 8004255);