#441108

order	512
nilpotency class	3
isoclinism	isoclinic class of #420461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,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, 348304);