#8108782

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	#40006	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,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,e^b=egi

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, 8015978);