#8933873

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	#51946	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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)(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)(33,34,39,41)(35,43,46,40)(36,37,45,47)(38,48,44,42);(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)(33,35,39,46)(34,40,41,43)(36,38,45,44)(37,42,47,48);(2,4)(3,8)(6,14)(13,16)(19,27)(20,30)(23,31)(24,29)(34,41)(36,45)(38,44)(40,43);(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)(33,36,39,45)(34,37,41,47)(35,44,46,38)(40,48,43,42);(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)(34,36)(35,42)(38,40)(39,47)(41,45)(43,44)(46,48);(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)(33,38,39,44)(34,42,41,48)(35,45,46,36)(37,40,47,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,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,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, 8841069);