#8285578

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	#52602	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=h,e^2=h,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, 8192774);