#8739434

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3725 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)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,7,5)(2,8,9,10)(4,14,11,12)(6,13,16,15)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,9)(4,11)(8,10)(12,14)(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4)(2,6)(3,12)(5,14)(7,11)(8,15)(9,16)(10,13)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,5,7,3)(2,10,9,8)(4,14,11,12)(6,13,16,15)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32);(1,6)(2,11)(3,13)(4,9)(5,15)(7,16)(8,12)(10,14)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,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, 8646630);