#445996

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353192);