#7003553

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42796	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6910749);