#8137104

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	#43433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=gh,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^d=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, 8044300);