#8425650

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332846);