#8746578

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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)(3,7)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18,22,26)(19,21,25,23)(20,28,30,24)(27,29,32,31)(33,34,38,41)(35,42,37,39)(36,40,45,48)(43,47,44,46);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,18,22,26)(19,21,25,23)(20,24,30,28)(27,31,32,29)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(3,12)(7,8)(13,16)(14,15)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,29,28,31)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,31)(30,32)(33,36,38,45)(34,40,41,48)(35,43,37,44)(39,46,42,47);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,31,23,29)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,37,38,35)(34,39,41,42)(36,43,45,44)(40,47,48,46)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=gh,f^2=gi,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, 8653774);