#8933871

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	#48649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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)(3,5)(4,7)(6,12)(8,11)(9,14)(10,16)(13,15)(17,18,22,26)(19,23,27,31)(20,29,30,24)(21,28,32,25)(33,34,39,42)(35,43,47,38)(36,40,41,37)(44,48,45,46);(1,3,7,8)(2,5,4,11)(6,10,14,15)(9,13,12,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,35)(34,38)(36,44)(37,48)(39,47)(40,46)(41,45)(42,43);(2,4)(3,8)(6,14)(13,16)(19,27)(20,30)(23,31)(24,29)(33,36,39,41)(34,40,42,37)(35,44,47,45)(38,46,43,48);(1,4,7,2)(3,5,8,11)(6,12,14,9)(10,13,15,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,36)(34,40)(35,45)(37,42)(38,48)(39,41)(43,46)(44,47);(1,5)(2,8)(3,4)(6,13)(7,11)(9,15)(10,12)(14,16)(17,21)(18,25)(19,24)(20,31)(22,32)(23,30)(26,28)(27,29)(33,37,39,40)(34,41,42,36)(35,46,47,48)(38,44,43,45);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,31)(29,32)(33,38)(34,35)(36,48)(37,44)(39,43)(40,45)(41,46)(42,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^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,d^c=dgi,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, 8841067);