#8107647

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=gi,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014843);