#8202301

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34)(35,43)(36,45)(37,42)(38,39)(40,44)(41,46)(47,48);(2,6)(7,9)(8,13)(14,15)(19,27)(20,30)(23,32)(24,29)(34,39)(36,46)(40,43)(42,48);(4,12)(8,13)(10,16)(14,15)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31)(33,35,38,44)(34,40,39,43)(36,42,46,48)(37,45,47,41);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(33,36)(34,41)(35,42)(37,40)(38,46)(39,45)(43,47)(44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,36)(34,41)(35,42)(37,40)(38,46)(39,45)(43,47)(44,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,37)(34,42)(35,41)(36,40)(38,47)(39,48)(43,46)(44,45)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gh,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, 8109497);