#10201954

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	#56350	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34,39,41)(35,44,45,40)(36,46,43,38)(37,42,47,48);(2,7)(4,10)(9,13)(12,14)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,35)(34,40)(36,37)(38,42)(39,45)(41,44)(43,47)(46,48);(2,7)(3,8)(9,13)(11,16)(18,23)(19,25)(24,32)(26,27)(34,41)(35,45)(36,43)(42,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,14)(15,16)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,36,39,43)(34,38,41,46)(35,37,45,47)(40,42,44,48);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28)(33,37)(34,42)(35,43)(36,45)(38,44)(39,47)(40,46)(41,48);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(33,38)(34,43)(35,42)(36,41)(37,44)(39,46)(40,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,d^2=ghi,e^2=gh,f^2=hi,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, 10109150);