#8893583

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800779);