#8295580

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	#55320	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=gi,e^2=gh,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, 8202776);