#8928006

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

4141 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=h,e^2=gh,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, 8835202);