#9712036

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3875 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(18,20)(23,25)(24,28)(30,31)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(2,8)(4,9)(7,14)(10,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,20,22,18)(19,25,27,23)(21,24,29,28)(26,30,32,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37,38,34)(35,41,43,39)(36,40,44,45)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9619232);