#10247201

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	#42017	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(3,9)(4,6)(5,8)(7,11)(10,15)(12,13)(14,16)(33,34,39,40)(35,45,47,41)(36,46,48,42)(37,43,44,38);(1,3,7,5)(2,8,11,9)(4,14,15,12)(6,13,10,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(34,40)(36,48)(37,44)(41,45);(1,3,7,5)(2,9,11,8)(4,12,15,14)(6,13,10,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35,39,47)(34,41,40,45)(36,44,48,37)(38,46,43,42);(1,4)(2,10)(3,12)(5,14)(6,11)(7,15)(8,13)(9,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,36,39,48)(34,42,40,46)(35,37,47,44)(38,41,43,45);(1,5,7,3)(2,8,11,9)(4,12,15,14)(6,13,10,16)(17,19)(18,20)(21,29)(22,27)(23,25)(24,31)(26,30)(28,32)(33,37)(34,43)(35,36)(38,40)(39,44)(41,42)(45,46)(47,48);(1,6,7,10)(2,4,11,15)(3,13,5,16)(8,14,9,12)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38,39,43)(34,44,40,37)(35,46,47,42)(36,41,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=hi,e^2=gh,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, 10154397);