#8927330

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

3990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,e^2=g,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, 8834526);