#8107600

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

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

Polycyclic presentation:

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

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, 8014796);