#8436623

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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)(3,10)(4,9)(5,14)(6,8)(7,11)(12,16)(13,15)(17,18,22,26)(19,21,25,23)(20,24,29,32)(27,28,31,30)(33,34,38,42)(35,43,44,39)(36,40,45,47)(37,41,46,48);(1,3,7,6)(2,8,11,10)(4,12,5,13)(9,15,14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(35,44)(36,45)(39,43)(40,47);(1,4,7,5)(2,9,11,14)(3,12,6,13)(8,15,10,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,41)(37,40)(38,44)(42,43)(45,48)(46,47);(1,5,7,4)(2,9,11,14)(3,13,6,12)(8,15,10,16)(17,18)(19,21)(20,24)(22,26)(23,25)(27,28)(29,32)(30,31)(33,36,38,45)(34,40,42,47)(35,41,44,48)(37,43,46,39);(1,6,7,3)(2,10,11,8)(4,13,5,12)(9,16,14,15)(17,20,22,29)(18,24,26,32)(19,27,25,31)(21,28,23,30);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,21)(18,25)(19,26)(20,28)(22,23)(24,31)(27,32)(29,30)(33,37)(34,41)(35,40)(36,39)(38,46)(42,48)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=gh,e^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, 8343819);