#10142158

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	#36731	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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)(19,25,27,22)(20,24,30,29)(26,32,28,31)(33,34,40,38)(35,44,46,41)(36,42,37,47)(39,43,45,48);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31)(33,35,40,46)(34,41,38,44)(36,45,37,39)(42,43,47,48);(2,7)(3,9)(8,16)(10,11)(18,23)(22,25)(24,29)(31,32)(33,36,40,37)(34,42,38,47)(35,45,46,39)(41,43,44,48);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,18)(19,22)(20,29)(21,23)(24,30)(25,27)(26,32)(28,31)(33,37,40,36)(34,42,38,47)(35,39,46,45)(41,43,44,48);(1,4,6,13)(2,8,7,16)(3,10,9,11)(5,14,15,12)(17,20)(18,24)(19,26)(21,30)(22,31)(23,29)(25,32)(27,28)(33,38,40,34)(35,44,46,41)(36,47,37,42)(39,43,45,48);(1,5)(2,3)(4,12)(6,15)(7,9)(8,11)(10,16)(13,14)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31)(33,39)(34,43)(35,36)(37,46)(38,48)(40,45)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,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, 10049354);