#8095161

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3856 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=h,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, 8002357);