#6817223

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24250	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2006 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:

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

Polycyclic presentation:

2^9|c^2=fgh,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724419);