#8813972

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	#52042	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721168);