#8720408

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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)(3,11,5,8)(4,14,15,10)(6,12,16,13)(17,18,23,24)(19,22,25,32)(20,21,29,30)(26,31,28,27)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3,7,5)(2,8,9,11)(4,12,15,13)(6,10,16,14)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)(34,39)(35,42)(41,44)(43,48);(2,9)(4,15)(8,11)(12,13)(18,24)(19,25)(20,29)(27,31)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,11,16,8)(17,20)(18,21)(19,26)(22,31)(23,29)(24,30)(25,28)(27,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,7,3)(2,11,9,8)(4,12,15,13)(6,10,16,14)(17,21)(18,20)(19,27)(22,28)(23,30)(24,29)(25,31)(26,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,6,7,16)(2,12,9,13)(3,10,5,14)(4,8,15,11)(17,22,23,32)(18,25,24,19)(20,27,29,31)(21,26,30,28)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627604);