#8095583

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	#45592	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=i,e^2=hi,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, 8002779);