#8095271

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=gh,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, 8002467);