#8096509

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×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=hi,f^2=hi,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, 8003705);