#8750742

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

3749 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)(3,11)(4,10)(5,8)(6,12)(7,9)(13,16)(14,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34)(35,39)(36,42)(37,45)(38,41)(40,46)(43,48)(44,47);(1,3,7,5)(2,8,9,11)(4,12,14,13)(6,10,16,15)(18,19)(23,25)(24,26)(30,31)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(2,9)(4,14)(8,11)(12,13)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4,7,14)(2,10,9,15)(3,13,5,12)(6,11,16,8)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,5,7,3)(2,11,9,8)(4,12,14,13)(6,10,16,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48);(1,6,7,16)(2,12,9,13)(3,10,5,15)(4,8,14,11)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gi,e^2=g,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, 8657938);