#8463753

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

3898 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,e^2=ghi,f^2=hi,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, 8370949);