#8295535

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

3961 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202731);