#10349092

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10256288);