#10247567

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154763);