#8425433

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	#40689	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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=h,e^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, 8332629);