#8393409

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	#48705	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,22,23)(19,27,29,24)(20,28,30,25)(21,26,32,31)(33,34,38,39)(35,37,41,47)(36,43,46,40)(42,48,44,45);(2,3)(4,11)(8,10)(12,16)(18,23)(20,30)(24,27)(26,31)(33,34,38,39)(35,37,41,47)(36,43,46,40)(42,48,44,45);(5,13)(8,10)(12,16)(14,15)(18,23)(21,32)(24,27)(25,28)(34,39)(37,47)(40,43)(45,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19,22,29)(18,24,23,27)(20,31,30,26)(21,28,32,25)(33,35,38,41)(34,37,39,47)(36,44,46,42)(40,48,43,45);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,25)(19,26)(21,27)(22,30)(23,28)(24,32)(29,31)(33,36,38,46)(34,40,39,43)(35,42,41,44)(37,45,47,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,32)(18,26,23,31)(19,28,29,25)(20,27,30,24)(33,37)(34,41)(35,39)(36,45)(38,47)(40,44)(42,43)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300605);