#8425499

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	#52324	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=g,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^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332695);