#10521307

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(3,10)(4,8)(5,7)(6,9)(11,15)(12,14)(13,16)(17,18,23,25)(19,26,20,24)(21,22,31,32)(27,30,29,28)(33,34,38,41)(35,42,36,39)(37,45,46,40)(43,47,44,48);(1,3,6,5)(2,7,9,10)(4,12,13,11)(8,15,16,14)(33,34,38,41)(35,42,36,39)(37,40,46,45)(43,48,44,47);(17,19,23,20)(18,24,25,26)(21,27,31,29)(22,28,32,30)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,4,6,13)(2,8,9,16)(3,11,5,12)(7,14,10,15)(17,20,23,19)(18,24,25,26)(21,27,31,29)(22,30,32,28)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,21)(18,22)(19,27)(20,29)(23,31)(24,28)(25,32)(26,30)(33,36,38,35)(34,39,41,42)(37,43,46,44)(40,48,45,47);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,16,15)(17,22)(18,21)(19,28)(20,30)(23,32)(24,27)(25,31)(26,29)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=ghi,e^2=i,f^2=g,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, 10428503);