#8898061

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

4063 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 8805257);