#10360921

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	#38214	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10268117);