#8772508

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=g,e^2=gh,f^2=h,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, 8679704);