#8445121

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

3935 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(20,28)(24,31)(27,32)(29,30)(33,34,39,38)(35,40,46,45)(36,42,37,41)(43,48,44,47);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,35,39,46)(34,40,38,45)(36,43,37,44)(41,47,42,48);(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,39,37)(34,41,38,42)(35,43,46,44)(40,47,45,48);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,39,36)(34,42,38,41)(35,44,46,43)(40,48,45,47);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38,39,34)(35,45,46,40)(36,42,37,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=g,e^2=ghi,f^2=gh,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, 8352317);