#8417465

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

3726 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(18,22)(20,27)(23,28)(26,31)(33,34,39,42)(35,37,40,36)(38,46,47,41)(43,44,48,45);(1,3)(2,8)(4,6)(5,15)(7,12)(9,13)(10,14)(11,16)(19,25)(23,28)(26,31)(30,32)(33,35,39,40)(34,36,42,37)(38,43,47,48)(41,44,46,45);(3,12)(6,13)(8,16)(10,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,48,46,43);(1,4)(2,5)(3,13)(6,12)(7,9)(8,10)(11,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,37,39,36)(34,35,42,40)(38,45,47,44)(41,43,46,48);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,15)(13,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46);(1,6,7,13)(2,10,11,15)(3,4,12,9)(5,8,14,16)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324661);