#8179674

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	#42471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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)(3,7)(4,9)(5,13)(6,8)(10,14)(11,12)(15,16)(18,20)(19,25)(24,28)(27,31)(33,34,39,38)(35,42,40,36)(37,45,46,41)(43,44,47,48);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35,39,40)(34,36,38,42)(37,43,46,47)(41,44,45,48);(2,8)(3,9)(10,13)(11,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,36,39,42)(34,40,38,35)(37,44,46,48)(41,47,45,43);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,37,39,46)(34,41,38,45)(35,43,40,47)(36,44,42,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,38,39,34)(35,42,40,36)(37,45,46,41)(43,48,47,44)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8086870);