#8127548

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	#56349	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(3,7)(4,11)(5,13)(6,9)(8,14)(10,12)(15,16)(18,22)(23,28)(24,29)(30,32)(33,34,38,42)(35,43,45,39)(36,40,46,48)(37,47,44,41);(1,3,6,11)(2,7,9,4)(5,10,14,16)(8,15,13,12)(19,25)(23,28)(26,31)(30,32)(33,35,38,45)(34,39,42,43)(36,44,46,37)(40,47,48,41);(1,2,6,9)(3,4,11,7)(5,8,14,13)(10,12,16,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,38,46)(34,40,42,48)(35,44,45,37)(39,47,43,41);(1,2)(3,4)(5,8)(6,9)(7,11)(10,12)(13,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,37,38,44)(34,41,42,47)(35,36,45,46)(39,40,43,48);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36)(34,40)(35,44)(37,45)(38,46)(39,47)(41,43)(42,48);(1,5,6,14)(2,8,9,13)(3,10,11,16)(4,12,7,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034744);