#8471111

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

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=gi,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, 8378307);