#9815185

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	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=ghi,e^2=g,f^2=gh,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, 9722381);