#8119550

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3892 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8026746);