#6994037

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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2161 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)(13,14,19,17)(15,23,25,36)(16,21,28,27)(18,31,33,22)(20,32,30,26)(24,37,39,44)(29,41,42,35)(34,43,40,38);(5,6)(7,11)(8,10)(9,12)(13,15,19,25)(14,20,17,30)(16,24,28,39)(18,32,33,26)(21,34,27,40)(22,36,31,23)(29,43,42,38)(35,44,41,37);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,24)(17,27)(18,29)(19,28)(20,34)(22,35)(23,37)(25,39)(26,38)(30,40)(31,41)(32,43)(33,42)(36,44);(7,12)(9,11)(13,17,19,14)(15,23,25,36)(16,27,28,21)(18,22,33,31)(20,32,30,26)(24,37,39,44)(29,35,42,41)(34,43,40,38);(6,10)(9,11)(15,25)(16,28)(20,30)(21,27)(23,36)(26,32)(29,42)(35,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,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, 6901233);