#8096794

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=g,e^2=gh,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, 8003990);