#8461614

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8368810);