#8125472

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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)(3,4,10,8)(5,13,14,9)(11,15,16,12)(17,18,23,20)(19,24,26,21)(22,28,30,25)(27,29,32,31)(34,38)(36,43)(39,44)(41,47);(2,7)(3,10)(9,13)(11,16)(17,19,23,26)(18,24,20,21)(22,27,30,32)(25,31,28,29)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48);(2,7)(4,8)(9,13)(12,15)(19,26)(21,24)(27,32)(29,31)(33,35,37,42)(34,39,38,44)(36,41,43,47)(40,46,45,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,35)(34,39)(36,41)(37,42)(38,44)(40,46)(43,47)(45,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,36,37,43)(34,40,38,45)(35,41,42,47)(39,46,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=i,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8032668);