#8741690

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=h,e^2=g,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, 8648886);