#8158975

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066171);