#9054434

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	#55717	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961630);