#8094654

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

3860 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)(3,6)(4,9)(5,15)(7,10)(8,13)(11,16)(12,14)(20,28)(24,29)(26,32)(30,31)(33,34,38,42)(35,39,46,43)(36,40,37,41)(44,47,45,48);(1,3)(2,8)(4,11)(5,12)(6,10)(7,13)(9,14)(15,16)(18,22)(23,25)(24,29)(30,31)(33,34,38,42)(35,43,46,39)(36,40,37,41)(44,48,45,47);(1,4,7,5)(2,9,10,15)(3,11,13,12)(6,16,8,14)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,7,4)(2,9,10,15)(3,12,13,11)(6,16,8,14)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,48)(42,43);(1,6,7,8)(2,3,10,13)(4,14,5,16)(9,12,15,11)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,3)(2,8)(4,11)(5,12)(6,10)(7,13)(9,14)(15,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8001850);