#9726385

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	#37214	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,16)(14,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,41)(36,40)(37,39)(38,42)(43,47)(44,46)(45,48);(1,3)(2,8)(4,14)(5,11)(6,15)(7,16)(9,10)(12,13)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(18,20)(19,26)(25,28)(27,29)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(1,4,7,11)(2,9,12,6)(3,14,16,5)(8,10,13,15)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,7,14)(2,10,12,15)(3,11,16,4)(6,8,9,13)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,6,7,9)(2,11,12,4)(3,15,16,10)(5,13,14,8)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,37,38,35)(34,41,42,39)(36,43,45,44)(40,46,48,47)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,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, 9633581);