#8186965

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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)(3,4)(5,8)(6,14)(7,10)(9,15)(11,13)(12,16)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31)(33,34,38,41)(35,39,42,36)(37,40,46,48)(43,47,45,44);(1,3,7,8)(2,4,10,5)(6,11,15,16)(9,12,14,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(35,42)(36,39)(43,45)(44,47);(1,4,7,5)(2,8,10,3)(6,12,15,13)(9,16,14,11)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,35)(34,39)(36,41)(37,45)(38,42)(40,44)(43,46)(47,48);(1,5,7,4)(2,8,10,3)(6,13,15,12)(9,16,14,11)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,36,38,39)(34,35,41,42)(37,44,46,47)(40,43,48,45);(1,6,7,15)(2,9,10,14)(3,11,8,16)(4,12,5,13)(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,48)(42,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gi,e^2=h,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, 8094161);