#8932827

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=gh,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, 8840023);