#8227321

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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:

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

Polycyclic presentation:

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

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, 8134517);