#8908406

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815602);