#8739293

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	#2819	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8646489);