#8688412

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	#42529	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595608);