#9912876

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	#55356	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3887 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=g,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, 9820072);