#8898912

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×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3934 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)(3,5)(4,9)(6,15)(7,8)(10,14)(11,12)(13,16)(17,18,23,28)(19,25,20,24)(21,26,31,29)(22,30,32,27)(34,36)(35,40)(38,46)(45,48);(1,3,7,6)(2,5,8,15)(4,10,13,12)(9,14,16,11)(17,19,23,20)(18,24,28,25)(21,30,31,27)(22,29,32,26)(33,34,39,36)(35,42,40,37)(38,41,46,44)(43,48,47,45);(2,8)(3,6)(9,16)(10,12)(17,19,23,20)(18,24,28,25)(21,27,31,30)(22,26,32,29)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(1,4)(2,9)(3,10)(5,14)(6,12)(7,13)(8,16)(11,15)(17,20,23,19)(18,25,28,24)(21,27,31,30)(22,26,32,29)(33,36,39,34)(35,42,40,37)(38,44,46,41)(43,48,47,45);(1,5)(2,3)(4,11)(6,8)(7,15)(9,12)(10,16)(13,14)(17,21,23,31)(18,26,28,29)(19,27,20,30)(22,24,32,25)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,6,7,3)(2,5,8,15)(4,12,13,10)(9,14,16,11)(17,22)(18,27)(19,29)(20,26)(21,24)(23,32)(25,31)(28,30)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=g,e^2=g,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, 8806108);