#10481051

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	#56931	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3853 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10388247);