#8097715

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	#45554	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004911);