#9851016

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9758212);