#8110413

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017609);