#9021957

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	#42102	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8929153);