#8496916

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	#40053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4206 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404112);