#8167384

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

4036 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=i,e^2=i,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^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, 8074580);