#8334580

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gi,e^2=gh,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, 8241776);