#8193820

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

4067 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8101016);