#8901379

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	#58904	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3983 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gi,e^2=ghi,f^2=g,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, 8808575);