#9924019

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	#55704	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3881 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831215);