#8179645

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

4051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,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, 8086841);