#8901570

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8808766);