#8137573

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	#49741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044769);