#9862983

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

3937 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,23,25)(19,24,22,32)(20,26,29,21)(27,28,30,31)(34,40)(36,45)(41,46)(43,44);(1,3,7,4)(2,8,9,10)(5,13,15,12)(6,14,16,11)(17,19,23,22)(18,24,25,32)(20,27,29,30)(21,31,26,28)(33,34)(35,41)(36,42)(37,46)(38,43)(39,40)(44,48)(45,47);(2,9)(3,4)(11,14)(12,13)(18,25)(20,29)(24,32)(27,30)(33,35,39,37)(34,41,40,46)(36,44,45,43)(38,42,48,47);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,11,16,14)(17,20)(18,26)(19,27)(21,25)(22,30)(23,29)(24,28)(31,32)(33,36)(34,42)(35,44)(37,43)(38,46)(39,45)(40,47)(41,48);(1,5)(2,11)(3,13)(4,12)(6,8)(7,15)(9,14)(10,16)(17,21,23,26)(18,20,25,29)(19,28,22,31)(24,30,32,27)(33,37,39,35)(34,41,40,46)(36,43,45,44)(38,42,48,47);(1,6,7,16)(2,12,9,13)(3,14,4,11)(5,10,15,8)(17,22,23,19)(18,24,25,32)(20,30,29,27)(21,31,26,28)(33,38,39,48)(34,43,40,44)(35,42,37,47)(36,46,45,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=i,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770179);