#8455680

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

3963 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,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, 8362876);