#8127538

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	#49934	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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)(3,8,5,11)(6,12,14,9)(10,16,13,15)(17,18,23,28)(19,27,22,24)(20,21,26,25)(29,32,31,30)(34,38)(35,41)(36,44)(46,48);(1,3,7,5)(2,8,4,11)(6,10,14,13)(9,15,12,16)(17,19,23,22)(18,24,28,27)(20,29,26,31)(21,30,25,32)(33,34,37,38)(35,39,41,42)(36,40,44,45)(43,46,47,48);(2,4)(3,5)(9,12)(10,13)(20,26)(21,25)(29,31)(30,32)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,4,7,2)(3,8,5,11)(6,12,14,9)(10,15,13,16)(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,32,27,30)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,5,7,3)(2,8,4,11)(6,13,14,10)(9,15,12,16)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,31,27,29)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,32,25,30)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=gh,e^2=gh,f^2=g,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, 8034734);