#9938131

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3888 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9845327);