#8701623

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8608819);