#8188183

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	#53134	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,10,5,8)(4,12,14,9)(11,15,13,16)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,28)(31,32)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3,7,5)(2,8,6,10)(4,11,14,13)(9,15,12,16)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(34,39)(35,42)(41,44)(43,48);(2,6)(8,10)(9,12)(15,16)(18,24)(19,25)(26,29)(27,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,5,7,3)(2,10,6,8)(4,13,14,11)(9,16,12,15)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,6,7,2)(3,10,5,8)(4,12,14,9)(11,16,13,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=i,e^2=h,f^2=g,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, 8095379);