#8942634

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8849830);