#6684764

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#25764	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2010 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)(5,6,8,9)(7,11,12,10)(13,14)(15,23)(16,28)(17,21)(18,22)(19,26)(20,30)(24,34)(25,36)(27,35)(29,37)(31,40)(32,39)(33,38)(41,42)(43,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,26)(14,19,21,23)(16,29,30,43)(18,25,32,27)(20,37,28,44)(22,35,39,36)(24,33,42,31)(34,40,41,38);(1,2,3,4)(6,9)(10,11)(13,16)(14,20)(15,24)(17,30)(18,33)(19,34)(21,28)(22,40)(23,41)(25,29)(26,42)(27,43)(31,32)(35,37)(36,44)(38,39);(5,7,8,12)(6,10,9,11)(14,21)(16,30)(19,23)(22,39)(24,42)(29,43)(31,33)(35,36);(15,25)(18,32)(19,35)(22,39)(23,36)(24,29)(26,27)(31,33)(34,37)(38,40)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6591960);