#8137632

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	#56401	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^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,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, 8044828);