#8161285

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	#41589	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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=gh,f^2=g,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^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, 8068481);