#8188512

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

3987 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,16)(13,14)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(1,3,7,5)(2,8,9,11)(4,12,15,14)(6,13,10,16)(18,20)(19,25)(24,28)(27,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(2,9)(6,10)(8,11)(13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,16)(11,13)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,16,10,13)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,12,11,14)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48)

Polycyclic presentation:

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

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, 8095708);