#8436702

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=g,e^2=gh,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, 8343898);