#8445848

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=gh,e^2=i,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, 8353044);