#8165194

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

4072 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8072390);