#8276821

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

4083 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)(3,11,12,6)(4,13,16,9)(5,15,14,10)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(35,43)(37,48)(40,44)(41,46);(1,3,7,12)(2,6,8,11)(4,14,16,5)(9,10,13,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,34,39,42)(35,44,43,40)(36,47,45,38)(37,41,48,46);(2,8)(6,11)(9,13)(10,15)(19,26)(23,30)(27,31)(28,32)(33,35,39,43)(34,40,42,44)(36,37,45,48)(38,41,47,46);(1,4,7,16)(2,9,8,13)(3,5,12,14)(6,15,11,10)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36)(34,38)(35,37)(39,45)(40,41)(42,47)(43,48)(44,46);(1,5)(2,10)(3,4)(6,13)(7,14)(8,15)(9,11)(12,16)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37)(34,41)(35,45)(36,43)(38,44)(39,48)(40,47)(42,46);(1,6,7,11)(2,3,8,12)(4,15,16,10)(5,13,14,9)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,38)(34,36)(35,46)(37,44)(39,47)(40,48)(41,43)(42,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8184017);