#8810175

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

3839 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8717371);