#8893665

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

4056 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=h,f^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, 8800861);