#7054580

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	C₂×#3269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2095 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)(13,14,20,24)(15,25,28,36)(16,22,31,38)(17,18,23,19)(21,34,37,27)(26,39,43,44)(29,30,32,33)(35,41,40,42);(5,6)(7,11)(8,10)(9,12)(13,15,20,28)(14,21,24,37)(16,26,31,43)(17,25,23,36)(18,27,19,34)(22,35,38,40)(29,44,32,39)(30,41,33,42);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,26)(17,32)(18,33)(19,30)(20,31)(21,35)(23,29)(24,38)(25,39)(27,42)(28,43)(34,41)(36,44)(37,40);(7,12)(9,11)(13,17)(14,18)(15,21)(16,29)(19,24)(20,23)(22,30)(25,34)(26,40)(27,36)(28,37)(31,32)(33,38)(35,43)(39,42)(41,44);(6,10)(9,11)(13,18,20,19)(14,23,24,17)(15,27,28,34)(16,30,31,33)(21,36,37,25)(22,32,38,29)(26,41,43,42)(35,39,40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^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=eh,f^b=fi,d^c=dhi,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, 6961776);