#10247115

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	#46248	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154311);