#497607

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

772 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=ei,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, 404803);