#8260693

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	#43193	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18,22,23)(19,25,26,24)(20,29,30,21)(27,28,32,31)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(2,7)(4,8)(9,14)(12,15)(18,23)(19,26)(20,30)(28,31)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(5,13)(9,14)(10,16)(12,15)(18,23)(21,29)(24,25)(28,31)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,19,22,26)(18,24,23,25)(20,27,30,32)(21,28,29,31)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,20)(18,21)(19,27)(22,30)(23,29)(24,28)(25,31)(26,32)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,21)(18,20)(19,28)(22,29)(23,30)(24,27)(25,32)(26,31)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)

Polycyclic presentation:

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

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, 8167889);