#10181092

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,34,39,37)(35,42,36,40)(38,41,46,45)(43,48,44,47);(2,7)(3,9)(8,16)(10,11)(18,20)(19,26)(25,28)(27,32)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,4,6,13)(2,8,7,16)(3,10,9,11)(5,14,15,12)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,37,39,34)(35,42,36,40)(38,41,46,45)(43,47,44,48);(1,5)(2,3)(4,12)(6,15)(7,9)(8,11)(10,16)(13,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(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=hi,c^2=hi,d^2=ghi,e^2=gi,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, 10088288);