#8285312

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	#39520	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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8192508);