#8898051

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

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, 8805247);