#8862453

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8769649);