#10020809

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

4094 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)(3,5)(4,8)(6,7)(9,14)(10,13)(11,15)(12,16)(19,25)(23,26)(27,31)(30,32)(33,34,39,36)(35,42,37,40)(38,41,46,44)(43,48,45,47);(2,7)(3,9)(4,11)(12,13)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(2,7)(3,9)(4,11)(12,13)(18,22)(23,26)(24,28)(30,32)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(1,3,6,9)(2,5,7,14)(4,10,11,16)(8,13,15,12)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4,6,11)(2,8,7,15)(3,10,9,16)(5,13,14,12)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,37,39,35)(34,40,36,42)(38,43,46,45)(41,48,44,47);(1,5)(2,9)(3,7)(4,12)(6,14)(8,10)(11,13)(15,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

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

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, 9928005);