#8322232

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3992 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,e^2=h,f^2=gi,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, 8229428);