#6449178

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3384 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:

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

Polycyclic presentation:

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

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, 6356374);