#8127562

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	#56296	C₂
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)(3,7)(4,11)(5,13)(6,9)(8,14)(10,12)(15,16)(18,22)(23,28)(24,29)(30,32)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,3,6,11)(2,7,9,4)(5,10,14,16)(8,15,13,12)(19,25)(23,28)(26,31)(30,32)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(1,2,6,9)(3,4,11,7)(5,8,14,13)(10,12,16,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,2)(3,4)(5,8)(6,9)(7,11)(10,12)(13,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,5,6,14)(2,8,9,13)(3,10,11,16)(4,12,7,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034758);