#10529822

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	#46076	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3898 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)(3,10)(4,7)(5,8)(6,9)(11,15)(12,13)(14,16)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(33,34,38,41)(35,37,40,43)(36,39,46,48)(42,45,44,47);(1,3,6,8)(2,5,9,10)(4,12,14,11)(7,15,16,13)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35)(34,37)(36,44)(38,40)(39,47)(41,43)(42,46)(45,48);(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,48)(43,47);(1,4,6,14)(2,7,9,16)(3,11,8,12)(5,13,10,15)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,34)(35,43)(36,39)(37,40)(38,41)(42,47)(44,45)(46,48);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,15)(14,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,36)(34,39)(35,44)(37,47)(38,46)(40,42)(41,48)(43,45);(1,5)(2,8)(3,9)(4,13)(6,10)(7,12)(11,16)(14,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37,38,43)(34,40,41,35)(36,45,46,47)(39,44,48,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10437018);