#496493

order	512
nilpotency class	4
isoclinism	isoclinic class of #481373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

849 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)(3,7)(9,10,14,17,15,20,13,29,12,18,27,35,28,36,26,19)(11,21,24,31,25,38,23,32,22,16,30,33,40,34,39,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,33)(18,21)(19,32)(20,34)(26,39)(27,30)(28,40)(29,37)(31,35)(36,38);(2,5)(6,8)(10,17)(11,22)(13,27)(14,26)(15,28)(16,31)(18,35)(19,20)(21,33)(23,24)(29,36)(30,39)(32,38)(34,37);(1,3,4,7)(2,6,5,8)(10,18)(16,21)(17,35)(19,29)(20,36)(31,33)(32,37)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313749, #72313766, #72313801, …
maximal normal subgroup of	#72313749, #72313766, #72313801, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403689);