#8445879

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3896 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)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(17,18,23,27)(19,28,29,24)(20,25,30,32)(21,22,26,31)(34,39)(35,42)(36,44)(46,47);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(19,29)(20,30)(24,28)(25,32)(33,34)(35,40)(36,45)(37,42)(38,39)(41,44)(43,47)(46,48);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,19)(18,24)(20,26)(21,30)(22,25)(23,29)(27,28)(31,32)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,46,45,47);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,20,23,30)(18,25,27,32)(19,26,29,21)(22,28,31,24)(33,36)(34,41)(35,43)(37,47)(38,44)(39,45)(40,46)(42,48);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,21,23,26)(18,22,27,31)(19,30,29,20)(24,25,28,32);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,22)(18,26)(19,25)(20,24)(21,27)(23,31)(28,30)(29,32)(33,37,38,40)(34,42,39,35)(36,46,44,47)(41,43,45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353075);