#8175735

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8082931);