#8108728

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015924);