#8202382

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	#40540	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8109578);