#8893428

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

4180 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)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,13,16,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(2,9)(6,16)(8,10)(13,15)(19,26)(23,30)(27,31)(28,32)(33,35,38,36)(34,40,42,39)(37,43,45,44)(41,47,48,46);(1,4,7,3)(2,10,9,8)(5,12,11,14)(6,15,16,13)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36,38,35)(34,40,42,39)(37,43,45,44)(41,46,48,47);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,15,10,13)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,12)(10,14)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800624);