#8928033

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

4182 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)(3,8,11,15)(4,9,13,12)(5,10,16,14)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,32)(30,31)(33,34,39,37)(35,42,40,36)(38,45,47,41)(43,44,48,46);(2,7)(4,13)(8,15)(10,14)(17,19)(18,22)(20,21)(23,27)(24,30)(25,29)(26,32)(28,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(3,11)(5,16)(8,15)(10,14)(19,27)(20,28)(22,32)(24,29)(33,36,39,42)(34,40,37,35)(38,46,47,44)(41,43,45,48);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,20)(18,24)(19,21)(22,30)(23,28)(25,32)(26,29)(27,31)(33,36)(34,35)(37,40)(38,44)(39,42)(41,43)(45,48)(46,47);(1,4)(2,9)(3,10)(5,15)(6,13)(7,12)(8,16)(11,14)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,30)(29,32)(33,37,39,34)(35,42,40,36)(38,45,47,41)(43,46,48,44);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,22)(18,19)(20,30)(21,24)(23,32)(25,28)(26,27)(29,31)(33,38,39,47)(34,41,37,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gh,e^2=h,f^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, 8835229);