#10481033

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	#45917	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gi,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, 10388229);