#8188350

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

3978 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)(3,9,4,8)(5,13,14,10)(11,15,12,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,45)(36,42)(37,47)(38,43)(39,41)(40,44)(46,48);(1,3,7,4)(2,8,6,9)(5,11,14,12)(10,15,13,16)(18,22)(23,25)(24,29)(30,31)(33,35,40,39)(34,41,44,45)(36,46,37,43)(38,47,48,42);(2,6)(8,9)(10,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,40,37)(34,42,44,47)(35,46,39,43)(38,45,48,41);(1,4,7,3)(2,9,6,8)(5,12,14,11)(10,16,13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,40,36)(34,42,44,47)(35,46,39,43)(38,41,48,45);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38,40,48)(34,43,44,46)(35,47,39,42)(36,45,37,41);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,39,40,35)(34,41,44,45)(36,46,37,43)(38,42,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=gi,e^2=gh,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, 8095546);