#6682336

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589532);