#8115431

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3804 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)(3,5)(6,14)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34,39,37)(35,40,36,43)(38,45,46,41)(42,48,44,47);(1,2)(3,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,3,7,5)(2,8,4,11)(6,10,14,13)(9,15,12,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,4,7,2)(3,8,5,11)(6,12,14,9)(10,15,13,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,5,7,3)(2,8,4,11)(6,13,14,10)(9,15,12,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48);(1,6,7,14)(2,9,4,12)(3,10,5,13)(8,15,11,16)

Polycyclic presentation:

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

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, 8022627);