#8455694

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	#2416	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)(3,8)(4,10)(5,9)(6,11)(7,12)(13,14)(15,16)(17,18,22,26)(19,23,24,20)(21,25,30,29)(27,31,32,28)(34,38)(39,45)(40,46)(47,48);(1,3,7,5)(2,8,12,9)(4,15,11,13)(6,14,10,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(34,38)(35,41)(40,46)(42,43);(1,3,7,5)(2,9,12,8)(4,13,11,15)(6,14,10,16)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,34)(35,39)(36,40)(37,38)(41,45)(42,47)(43,48)(44,46);(1,4,7,11)(2,10,12,6)(3,13,5,15)(8,14,9,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,28)(31,32)(33,35,37,41)(34,39,38,45)(36,43,44,42)(40,48,46,47);(1,5,7,3)(2,8,12,9)(4,15,11,13)(6,16,10,14)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27);(1,6,7,10)(2,11,12,4)(3,14,5,16)(8,13,9,15)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)(33,36,37,44)(34,40,38,46)(35,42,41,43)(39,47,45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=h,e^2=ghi,f^2=gh,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, 8362890);