#6449053

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3392 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)(7,10)(8,9)(11,12)(13,14,19,24)(15,21,27,38)(16,28,17,22)(18,33,20,23)(25,36,43,35)(26,39,34,37)(29,44,31,40)(30,41,32,42);(1,2,3,4)(13,15)(14,21)(16,29)(17,31)(18,26)(19,27)(20,34)(22,40)(23,37)(24,38)(25,30)(28,44)(32,43)(33,39)(35,42)(36,41);(6,9)(10,12)(13,16,19,17)(14,22,24,28)(15,25,27,43)(18,30,20,32)(21,35,38,36)(23,41,33,42)(26,31,34,29)(37,40,39,44);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,22,24,28)(15,25,27,43)(18,32,20,30)(21,36,38,35)(23,41,33,42)(26,31,34,29)(37,44,39,40);(13,18,19,20)(14,23,24,33)(15,26,27,34)(16,30,17,32)(21,37,38,39)(22,41,28,42)(25,31,43,29)(35,40,36,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356249);