#8446022

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	#42111	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18,23,26)(19,27,20,24)(21,22,25,32)(28,31,30,29)(33,34,38,40)(35,41,43,37)(36,39,45,46)(42,47,48,44);(2,7)(3,10)(8,14)(11,16)(19,20)(24,27)(28,30)(29,31)(33,35,38,43)(34,37,40,41)(36,42,45,48)(39,44,46,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19,23,20)(18,24,26,27)(21,28,25,30)(22,29,32,31)(33,34)(35,37)(36,39)(38,40)(41,43)(42,44)(45,46)(47,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,20,23,19)(18,24,26,27)(21,30,25,28)(22,29,32,31)(33,34)(35,41)(36,39)(37,43)(38,40)(42,47)(44,48)(45,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21,23,25)(18,22,26,32)(19,28,20,30)(24,29,27,31)(33,36,38,45)(34,39,40,46)(35,42,43,48)(37,44,41,47);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,22)(18,25)(19,29)(20,31)(21,26)(23,32)(24,30)(27,28)(33,37,38,41)(34,35,40,43)(36,44,45,47)(39,42,46,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8353218);