#8228569

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,2,6,7)(3,5,9,14)(4,11,13,8)(10,16,15,12)(18,23)(19,26)(24,30)(27,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(2,7)(5,14)(8,11)(12,16)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=hi,e^2=hi,f^2=g,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, 8135765);