#8425448

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	#38543	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=h,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, 8332644);