#8118573

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	#39825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8025769);