#8464221

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	#42892	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,14)(15,16)(17,18)(19,24)(20,25)(21,31)(22,27)(23,28)(26,30)(29,32)(33,34,38,42)(35,39,37,41)(36,44,45,40)(43,48,46,47);(3,14)(4,15)(8,13)(9,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,3)(2,8)(4,11)(5,15)(6,16)(7,14)(9,10)(12,13)(17,19,23,32)(18,24,28,29)(20,30,27,21)(22,31,25,26)(33,35,38,37)(34,39,42,41)(36,43,45,46)(40,47,44,48);(1,4,7,15)(2,9,12,16)(3,11,14,5)(6,8,10,13)(17,20)(18,25)(19,30)(21,32)(22,28)(23,27)(24,26)(29,31)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,48)(42,44);(1,5,7,11)(2,10,12,6)(3,15,14,4)(8,9,13,16)(17,21,23,30)(18,26,28,31)(19,20,32,27)(22,29,25,24)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,6,7,10)(2,11,12,5)(3,16,14,9)(4,13,15,8)(17,22,23,25)(18,27,28,20)(19,31,32,26)(21,29,30,24)(33,37,38,35)(34,41,42,39)(36,43,45,46)(40,47,44,48)

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=h,e^2=i,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, 8371417);