#8821947

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

4064 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8729143);