#8927340

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

4300 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,23,24)(19,21,20,27)(22,30,31,25)(26,32,28,29)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(2,7)(4,13)(8,15)(10,14)(17,19,23,20)(18,21,24,27)(22,26,31,28)(25,29,30,32)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(3,11)(5,16)(8,15)(10,14)(18,24)(19,20)(22,31)(29,32)(34,36)(38,46)(40,42)(43,45);(1,3)(2,8)(4,14)(5,12)(6,11)(7,15)(9,16)(10,13)(17,20,23,19)(18,21,24,27)(22,28,31,26)(25,29,30,32)(33,36,39,34)(35,42,37,40)(38,44,46,41)(43,48,45,47);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,15,16,8)(17,21)(18,20)(19,24)(22,29)(23,27)(25,28)(26,30)(31,32)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,5)(2,10)(3,12)(4,15)(6,16)(7,14)(8,13)(9,11)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8834536);