#8295612

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	#57579	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8202808);