#8458912

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=ghi,f^2=gi,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, 8366108);