#8467218

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(17,18,24,22)(19,29,31,25)(20,27,21,26)(23,28,32,30)(33,34,39,38)(35,40,45,43)(36,41,46,48)(37,47,44,42);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,19,24,31)(18,25,22,29)(20,30,21,28)(23,27,32,26)(33,34,39,38)(35,43,45,40)(36,41,46,48)(37,42,44,47);(17,20,24,21)(18,26,22,27)(19,30,31,28)(23,29,32,25)(33,35,39,45)(34,40,38,43)(36,44,46,37)(41,42,48,47);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,21,24,20)(18,27,22,26)(19,30,31,28)(23,29,32,25)(33,36)(34,41)(35,44)(37,45)(38,48)(39,46)(40,42)(43,47);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,22,24,18)(19,29,31,25)(20,27,21,26)(23,30,32,28)(33,37,39,44)(34,42,38,47)(35,36,45,46)(40,48,43,41);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,23,24,32)(18,28,22,30)(19,27,31,26)(20,29,21,25)(33,38,39,34)(35,43,45,40)(36,41,46,48)(37,47,44,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=h,e^2=hi,f^2=gi,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, 8374414);