#8922795

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	#55498	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,18)(19,24)(20,30)(21,25)(22,26)(23,29)(27,28)(31,32)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,31)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(2,6)(4,13)(7,9)(10,16)(20,25)(21,30)(27,31)(28,32)(35,41)(36,44)(39,43)(40,45);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,22,25)(18,21,26,30)(19,27,29,31)(23,28,24,32)(33,36,38,44)(34,40,37,45)(35,42,41,47)(39,46,43,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,19)(18,24)(20,31)(21,28)(22,29)(23,26)(25,27)(30,32)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21)(18,25)(19,28)(20,26)(22,30)(23,31)(24,27)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=hi,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829991);