#8096845

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	#46042	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^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, 8004041);