#8202354

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	#48138	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3947 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=gh,e^2=g,f^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, 8109550);