#8188058

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	#45663	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095254);