#8334328

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	#44438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=gi,e^2=g,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, 8241524);