#8455691

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

3990 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,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, 8362887);