#8445895

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

3939 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)(5,7)(6,9)(8,12)(13,15)(14,16)(17,18,23,27)(19,28,29,24)(20,25,30,32)(21,22,26,31)(34,40)(35,44)(36,46)(43,48);(1,3,8,6)(2,9,12,11)(4,13,5,14)(7,15,10,16)(19,29)(20,30)(24,28)(25,32)(33,34)(35,41)(36,47)(37,43)(38,44)(39,40)(42,46)(45,48);(1,4,8,5)(2,10,12,7)(3,13,6,14)(9,16,11,15)(17,19)(18,24)(20,26)(21,30)(22,25)(23,29)(27,28)(31,32)(33,35,39,44)(34,41,40,38)(36,45,46,37)(42,43,47,48);(1,5,8,4)(2,10,12,7)(3,14,6,13)(9,16,11,15)(17,20,23,30)(18,25,27,32)(19,26,29,21)(22,28,31,24)(33,36)(34,42)(35,45)(37,44)(38,48)(39,46)(40,47)(41,43);(1,6,8,3)(2,11,12,9)(4,14,5,13)(7,16,10,15)(17,21,23,26)(18,22,27,31)(19,30,29,20)(24,25,28,32)(33,37,39,45)(34,43,40,48)(35,36,44,46)(38,42,41,47);(1,7,8,10)(2,5,12,4)(3,15,6,16)(9,14,11,13)(17,22)(18,26)(19,25)(20,24)(21,27)(23,31)(28,30)(29,32)(33,38,39,41)(34,44,40,35)(36,43,46,48)(37,42,45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8353091);