#8459125

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	#36211	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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)(3,13)(4,10)(5,7)(6,15)(8,12)(9,16)(11,14)(20,28)(24,31)(27,32)(29,30)(33,34,37,40)(35,38,43,41)(36,39,45,48)(42,46,44,47);(1,3)(2,9)(4,6)(5,14)(7,15)(8,16)(10,11)(12,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,4,8,5)(2,10,12,7)(3,6,16,14)(9,11,13,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,5,8,4)(2,10,12,7)(3,14,16,6)(9,11,13,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(1,6,8,14)(2,11,12,15)(3,4,16,5)(7,9,10,13)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30);(1,7,8,10)(2,5,12,4)(3,15,16,11)(6,9,14,13)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,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, 8366321);