#8324168

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,13)(14,16)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34,39,42)(35,38,41,47)(36,37,40,46)(43,44,45,48);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,13,10,16)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,35)(34,38)(36,43)(37,44)(39,41)(40,45)(42,47)(46,48);(2,9)(6,10)(8,11)(13,16)(18,23)(19,25)(24,32)(26,27)(35,41)(38,47)(43,45)(44,48);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,13)(11,16)(17,20)(18,24)(19,26)(21,30)(22,29)(23,32)(25,27)(28,31)(33,36)(34,37)(35,43)(38,44)(39,40)(41,45)(42,46)(47,48);(1,5,7,3)(2,11,9,8)(4,12,15,14)(6,16,10,13)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(33,37)(34,40)(35,44)(36,42)(38,45)(39,46)(41,48)(43,47);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,19)(18,21)(20,26)(22,25)(23,31)(24,30)(27,29)(28,32)(33,38,39,47)(34,41,42,35)(36,44,40,48)(37,45,46,43)

Polycyclic presentation:

2^9|a^2=i,b^2=g,e^2=g,f^2=hi,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, 8231364);