#497661

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂³	C₂²×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#557 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

789 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)(9,10,15,21,13,19,12,18)(11,22,24,34,23,16,26,31)(14,20,27,35,28,17,29,33)(25,32,38,39,36,30,37,40);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,26)(13,23)(14,25)(15,24)(17,32)(18,34)(19,22)(20,30)(21,31)(27,38)(28,36)(29,37)(33,39)(35,40);(1,3,5,4)(2,7,6,8)(10,17)(11,23)(14,28)(16,30)(18,33)(19,20)(21,35)(22,32)(24,26)(27,29)(31,39)(34,40);(1,4,5,3)(2,7,6,8)(9,12,13,15)(10,18,19,21)(11,24,23,26)(14,27,28,29)(16,31,22,34)(17,33,20,35)(25,37,36,38)(30,39,32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=ei,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404857);