#8425338

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	#38280	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4148 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=g,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, 8332534);