#8202275

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

3848 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)(3,11)(4,15)(5,16)(6,10)(7,8)(9,12)(13,14)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(2,8)(3,12)(4,16)(10,14)(19,27)(20,30)(23,32)(24,29)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,3,7,12)(2,9,8,11)(4,13,16,6)(5,14,15,10)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4)(2,5)(3,13)(6,12)(7,16)(8,15)(9,14)(10,11)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,5)(2,4)(3,14)(6,11)(7,15)(8,16)(9,13)(10,12)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,6,7,13)(2,10,8,14)(3,4,12,16)(5,11,15,9)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109471);