#8324355

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=h,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, 8231551);