#10183125

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	#48175	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 10090321);