#8095266

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	#45657	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002462);