#8456977

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	#52956	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,39,43)(35,40,46,44)(36,38,42,48)(37,47,45,41);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(18,22)(23,25)(24,29)(30,31)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,39,46)(34,40,43,44)(36,45,42,37)(38,41,48,47);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,38)(35,45)(37,46)(39,42)(40,41)(43,48)(44,47);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8364173);