#8471104

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(34,39)(35,42)(41,46)(43,48);(1,2,6,4)(3,7,9,10)(5,8,14,12)(11,15,13,16)(18,24)(19,20)(26,31)(27,28)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,46)(45,48);(1,3)(2,7)(4,10)(5,13)(6,9)(8,16)(11,14)(12,15)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,35)(34,37)(36,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,2)(3,10,9,7)(5,12,14,8)(11,16,13,15)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32)(33,36,38,45)(34,41,39,46)(35,43,42,48)(37,44,40,47);(1,5,6,14)(2,8,4,12)(3,11,9,13)(7,15,10,16)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,37)(34,42)(35,39)(36,44)(38,40)(41,48)(43,46)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=i,e^2=gi,f^2=ghi,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, 8378300);