#8621386

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	#55048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18)(19,24)(20,26)(21,30)(22,28)(23,25)(27,29)(31,32)(33,34,38,41)(35,42,37,39)(36,40,45,48)(43,47,44,46);(2,7)(3,11)(4,14)(13,15)(17,19)(18,24)(20,29)(21,31)(22,25)(23,28)(26,27)(30,32)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(2,7)(4,14)(8,10)(12,16)(18,25)(19,28)(21,29)(26,32)(36,45)(40,48)(43,44)(46,47);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,13)(14,16)(17,20)(18,26)(19,21)(22,27)(23,31)(24,30)(25,32)(28,29)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,21)(18,27)(19,20)(22,26)(23,29)(24,32)(25,30)(28,31)(33,36,38,45)(34,40,41,48)(35,43,37,44)(39,46,42,47);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,22)(18,28)(19,25)(20,30)(21,26)(23,24)(27,31)(29,32)(33,37,38,35)(34,39,41,42)(36,44,45,43)(40,46,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528582);