#8926159

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	#35508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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)(3,11)(4,10)(5,8)(6,14)(7,9)(12,13)(15,16)(19,25)(23,30)(26,28)(31,32)(33,34,39,42)(35,37,40,36)(38,46,47,41)(43,44,48,45);(1,3,7,5)(2,8,9,11)(4,12,14,15)(6,13,10,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(2,9)(6,10)(8,11)(13,16)(18,22)(23,30)(24,27)(31,32)(33,36,39,37)(34,40,42,35)(38,45,47,44)(41,43,46,48);(1,4)(2,10)(3,12)(5,15)(6,9)(7,14)(8,16)(11,13)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,5,7,3)(2,8,9,11)(4,12,14,15)(6,16,10,13)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37,39,36)(34,40,42,35)(38,45,47,44)(41,48,46,43);(1,6,7,10)(2,4,9,14)(3,13,5,16)(8,12,11,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,38,39,47)(34,41,42,46)(35,43,40,48)(36,44,37,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833355);