#10142156

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

4090 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10049352);