#8180385

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	#42071	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34)(35,36)(37,43)(38,40)(39,41)(42,44)(45,48)(46,47);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,41)(38,42)(39,43)(40,45)(44,47)(46,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,36,39,37)(34,35,41,43)(38,44,46,45)(40,42,47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,37,39,36)(34,35,41,43)(38,45,46,44)(40,42,47,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,38,39,46)(34,40,41,47)(35,42,43,48)(36,44,37,45);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,35)(34,36)(37,41)(38,42)(39,43)(40,44)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=g,e^2=gh,f^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, 8087581);