#8256623

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=gh,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8163819);