#8932733

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
prime radical	#46676	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839929);